Angular momentum

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Angular momentum
This gyroscope remains upright while spinning owing to the conservation of its angular momentum.

Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of an isolated system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.

The three-dimensional angular momentum for a point particle is classically represented as a pseudovector r × p, the cross product of the particle's position vector r (relative to some origin) and its momentum vector; the latter is p = mv in Newtonian mechanics. Unlike linear momentum, angular momentum depends on where this origin is chosen, since the particle's position is measured from it.

Angular momentum is an extensive quantity; that is, the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous rigid body or a fluid, the total angular momentum is the volume integral of angular momentum density (angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body.

Similar to conservation of linear momentum in the absence of external forces, angular momentum is conserved if there is no external torque. Torque can be defined as the rate of change of angular momentum, analogous to force. The net external torque on any system is always equal to the total torque on the system; the sum of all internal torques of any system is always 0 (this is the rotational analogue of Newton's third law of motion). Therefore, for an isolated system (where there is no net external torque), the total torque on the system must be 0, which means that the total angular momentum of the system is constant.

The change in angular momentum for a particular interaction is called angular impulse, sometimes twirl. Angular impulse is the angular analog of (linear) impulse.

Examples

The trivial case of the angular momentum ${\displaystyle L}$ of a body in an orbit is given by $${\displaystyle L=2\pi Mf{r}^2}$$ where ${\displaystyle M}$ is the mass of the orbiting object, ${\displaystyle f}$ is the orbit's frequency and ${\displaystyle r}$ is the orbit's radius.

The angular momentum ${\displaystyle L}$ of a uniform rigid sphere rotating around its axis, instead, is given by $${\displaystyle L=\frac{4}{5}\pi Mf{r}^2}$$ where ${\displaystyle M}$ is the sphere's mass, ${\displaystyle f}$ is the frequency of rotation and ${\displaystyle r}$ is the sphere's radius.

Thus, for example, the orbital angular momentum of the Earth with respect to the Sun is about 2.66 × 10⁴⁰ J⋅s, while its rotational angular momentum is about 7.05 × 10³³ J⋅s.

In the case of a uniform rigid sphere rotating around its axis, if, instead of its mass, its density is known, the angular momentum ${\displaystyle L}$ is given by $${\displaystyle L=\frac{16}{15}{\pi }^2\rho f{r}^5}$$ where ${\displaystyle \rho }$ is the sphere's density, ${\displaystyle f}$ is the frequency of rotation and ${\displaystyle r}$ is the sphere's radius.

In the simplest case of a spinning disk, the angular momentum ${\displaystyle L}$ is given by $${\displaystyle L=\pi Mf{r}^2}$$ where ${\displaystyle M}$ is the disk's mass, ${\displaystyle f}$ is the frequency of rotation and ${\displaystyle r}$ is the disk's radius.

If instead the disk rotates about its diameter (e.g. coin toss), its angular momentum ${\displaystyle L}$ is given by $${\displaystyle L=\frac{1}{2}\pi Mf{r}^2}$$

Definition in classical mechanics

"Orbital angular momentum" redirects here. For other uses, see Orbital angular momentum (disambiguation).

"Spin angular momentum" redirects here. For other uses, see Spin angular momentum (disambiguation).

Just as for angular velocity, there are two special types of angular momentum of an object: the spin angular momentum is the angular momentum about the object's center of mass, while the orbital angular momentum is the angular momentum about a chosen center of rotation. The Earth has an orbital angular momentum by nature of revolving around the Sun, and a spin angular momentum by nature of its daily rotation around the polar axis. The total angular momentum is the sum of the spin and orbital angular momenta. In the case of the Earth the primary conserved quantity is the total angular momentum of the Solar System because angular momentum is exchanged to a small but important extent among the planets and the Sun. The orbital angular momentum vector of a point particle is always parallel and directly proportional to its orbital angular velocity vector ω, where the constant of proportionality depends on both the mass of the particle and its distance from origin. The spin angular momentum vector of a rigid body is proportional but not always parallel to the spin angular velocity vector Ω, making the constant of proportionality a second-rank tensor rather than a scalar.

Orbital angular momentum in two dimensions

Velocity of the particle m with respect to the origin O can be resolved into components parallel to (v_∥) and perpendicular to (v_⊥) the radius vector r. The angular momentum of m is proportional to the perpendicular component v_⊥ of the velocity, or equivalently, to the perpendicular distance r_⊥ from the origin.

Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar). Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum p is proportional to mass m and linear speed v, $${\displaystyle p=mv,}$$ angular momentum L is proportional to moment of inertia I and angular speed ω measured in radians per second. $${\displaystyle L=I\omega .}$$

Unlike mass, which depends only on amount of matter, moment of inertia depends also on the position of the axis of rotation and the distribution of the matter. Unlike linear velocity, which does not depend upon the choice of origin, orbital angular velocity is always measured with respect to a fixed origin. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center.

In the case of circular motion of a single particle, we can use ${\displaystyle I=r^{2}m}$ and ${\displaystyle \omega =v/r}$ to expand angular momentum as ${\displaystyle L=r^{2}m\cdot v/r,}$ reducing to: $${\displaystyle L=rmv,}$$

the product of the radius of rotation r and the linear momentum of the particle ${\displaystyle p=mv}$, where ${\displaystyle v=r\omega }$ is the linear (tangential) speed.

This simple analysis can also apply to non-circular motion if one uses the component of the motion perpendicular to the radius vector: $${\displaystyle L=rm{v}_{\perp },}$$ where ${\displaystyle v_{\perp }=v\sin (\theta )}$ is the perpendicular component of the motion. Expanding, ${\displaystyle L=rmv\sin (\theta ),}$ rearranging, ${\displaystyle L=r\sin (\theta )mv,}$ and reducing, angular momentum can also be expressed, $${\displaystyle L=r_{\perp }mv,}$$ where ${\displaystyle r_{\perp }=r\sin (\theta )}$ is the length of the moment arm, a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, (length of moment arm) × (linear momentum), to which the term moment of momentum refers.

Scalar angular momentum from Lagrangian mechanics

Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum) of the angular coordinate ${\displaystyle \varphi }$ expressed in the Lagrangian of the mechanical system. Consider a mechanical system with a mass ${\displaystyle m}$ constrained to move in a circle of radius ${\displaystyle r}$ in the absence of any external force field. The kinetic energy of the system is $${\displaystyle T=\frac{1}{2}m{r}^2{\omega }^2=\frac{1}{2}m{r}^2{\dot{\varphi }}^2.}$$

And the potential energy is $${\displaystyle U=0.}$$

Then the Lagrangian is $${\displaystyle \mathcal{L}\left(\varphi ,\dot{\varphi }\right)=T-U=\frac{1}{2}m{r}^2{\dot{\varphi }}^2.}$$

The generalized momentum "canonically conjugate to" the coordinate ${\displaystyle \varphi }$ is defined by $${\displaystyle p_{\varphi }=\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }\dot{\varphi }}=m{r}^2\dot{\varphi }=I\omega =L.}$$

Orbital angular momentum in three dimensions

Relationship between force (F), torque (τ), momentum (p), and angular momentum (L) vectors in a rotating system. r is the position vector.

To completely define orbital angular momentum in three dimensions, it is required to know the rate at which the position vector sweeps out an angle, the direction perpendicular to the instantaneous plane of angular displacement, and the mass involved, as well as how this mass is distributed in space. By retaining this vector nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional motion about the center of rotation – circular, linear, or otherwise. In vector notation, the orbital angular momentum of a point particle in motion about the origin can be expressed as: $${\displaystyle \mathbf{L}=I\mathit{\omega },}$$ where

• ${\displaystyle I=r^{2}m}$ is the moment of inertia for a point mass,
• ${\displaystyle \mathit{\omega }=\frac{\mathbf{r}\times \mathbf{v}}{r^2}}$ is the orbital angular velocity of the particle about the origin,
• ${\displaystyle \mathbf{r}}$ is the position vector of the particle relative to the origin, and ${\displaystyle r=\left|\mathbf{r}\right|}$,
• ${\displaystyle \mathbf{v}}$ is the linear velocity of the particle relative to the origin, and
• ${\displaystyle m}$ is the mass of the particle.

This can be expanded, reduced, and by the rules of vector algebra, rearranged: $${\displaystyle \begin{array}{rl}\mathbf{L}& =\left(r^{2}m\right)\left(\frac{\mathbf{r}\times \mathbf{v}}{r^2}\right)\\ & =m\left(\mathbf{r}\times \mathbf{v}\right)\\ & =\mathbf{r}\times m\mathbf{v}\\ & =\mathbf{r}\times \mathbf{p},\end{array}}$$ which is the cross product of the position vector ${\displaystyle \mathbf{r}}$ and the linear momentum ${\displaystyle \mathbf{p}=m\mathbf{v}}$ of the particle. By the definition of the cross product, the ${\displaystyle \mathbf{L}}$ vector is perpendicular to both ${\displaystyle \mathbf{r}}$ and ${\displaystyle \mathbf{p}}$. It is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the ${\displaystyle \mathbf{L}}$ vector defines the plane in which ${\displaystyle \mathbf{r}}$ and ${\displaystyle \mathbf{p}}$ lie.

By defining a unit vector ${\displaystyle \hat{\mathbf{u}}}$ perpendicular to the plane of angular displacement, a scalar angular speed ${\displaystyle \omega }$ results, where $${\displaystyle \omega \hat{\mathbf{u}}=\mathit{\omega },}$$ and $${\displaystyle \omega =\frac{v_{\perp }}{r},}$$ where ${\displaystyle v_{\perp }}$ is the perpendicular component of the motion, as above.

The two-dimensional scalar equations of the previous section can thus be given direction: $${\displaystyle \begin{array}{rl}\mathbf{L}& =I\mathit{\omega }\\ & =I\omega \hat{\mathbf{u}}\\ & =\left(r^{2}m\right)\omega \hat{\mathbf{u}}\\ & =rm{v}_{\perp }\hat{\mathbf{u}}\\ & =r_{\perp }mv\hat{\mathbf{u}},\end{array}}$$ and ${\displaystyle \mathbf{L}=rmv\hat{\mathbf{u}}}$ for circular motion, where all of the motion is perpendicular to the radius ${\displaystyle r}$.

In the spherical coordinate system the angular momentum vector expresses as

${\displaystyle \mathbf{L}=m\mathbf{r}\times \mathbf{v}=m{r}^2\left(\dot{\theta }\hat{\mathit{\phi }}-\dot{\phi }\sin \theta \hat{\mathit{\theta }}\right).}$

Analogy to linear momentum

Angular momentum can be described as the rotational analog of linear momentum. Like linear momentum it involves elements of mass and displacement. Unlike linear momentum it also involves elements of position and shape.

Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy, the ability to do work, can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass, and displacement by its velocity. Their product, $${\displaystyle \begin{array}{rl}(\text{amount of inertia})\times (\text{amount of displacement})& =\text{amount of (inertia⋅displacement)}\\ \text{mass}\times \text{velocity}& =\text{momentum}\\ m\times v& =p\end{array}}$$ is the matter's momentum. Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the moment arm. It has the effect of multiplying the momentum's effort in proportion to its length, an effect known as a moment. Hence, the particle's momentum referred to a particular point, $${\displaystyle \begin{array}{rl}(\text{moment arm})\times (\text{amount of inertia})\times (\text{amount of displacement})& =\text{moment of (inertia⋅displacement)}\\ \text{length}\times \text{mass}\times \text{velocity}& =\text{moment of momentum}\\ r\times m\times v& =L\end{array}}$$ is the angular momentum, sometimes called, as here, the moment of momentum of the particle versus that particular center point. The equation ${\displaystyle L=rmv}$ combines a moment (a mass ${\displaystyle m}$ turning moment arm ${\displaystyle r}$) with a linear (straight-line equivalent) speed ${\displaystyle v}$. Linear speed referred to the central point is simply the product of the distance ${\displaystyle r}$ and the angular speed ${\displaystyle \omega }$ versus the point: ${\displaystyle v=r\omega ,}$ another moment. Hence, angular momentum contains a double moment: ${\displaystyle L=rmr\omega .}$ Simplifying slightly, ${\displaystyle L=r^{2}m\omega ,}$ the quantity ${\displaystyle r^{2}m}$ is the particle's moment of inertia, sometimes called the second moment of mass. It is a measure of rotational inertia.

Moment of inertia (shown here), and therefore angular momentum, is different for each shown configuration of mass and axis of rotation.

The above analogy of the translational momentum and rotational momentum can be expressed in vector form:

• $\mathbf{p}=m\mathbf{v}$ for linear motion
• ${\displaystyle \mathbf{L}=I\mathit{\omega }}$ for rotation

The direction of momentum is related to the direction of the velocity for linear movement. The direction of angular momentum is related to the angular velocity of the rotation.

Because moment of inertia is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the squares of their distances from the center of rotation. Therefore, the total moment of inertia, and the angular momentum, is a complex function of the configuration of the matter about the center of rotation and the orientation of the rotation for the various bits.

For a rigid body, for instance a wheel or an asteroid, the orientation of rotation is simply the position of the rotation axis versus the matter of the body. It may or may not pass through the center of mass, or it may lie completely outside of the body. For the same body, angular momentum may take a different value for every possible axis about which rotation may take place. It reaches a minimum when the axis passes through the center of mass.

For a collection of objects revolving about a center, for instance all of the bodies of the Solar System, the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to the system's axis. Their orientations may also be completely random.

In brief, the more mass and the farther it is from the center of rotation (the longer the moment arm), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity. In many cases the moment of inertia, and hence the angular momentum, can be simplified by, $${\displaystyle I=k^{2}m,}$$where ${\displaystyle k}$ is the radius of gyration, the distance from the axis at which the entire mass ${\displaystyle m}$ may be considered as concentrated.

Similarly, for a point mass ${\displaystyle m}$ the moment of inertia is defined as, $${\displaystyle I=r^{2}m}$$where ${\displaystyle r}$ is the radius of the point mass from the center of rotation, and for any collection of particles ${\displaystyle m_i}$ as the sum, $${\displaystyle \sum _i{I}_i=\sum _i{r}_i^2{m}_i.}$$

Angular momentum's dependence on position and shape is reflected in its units versus linear momentum: kg⋅m²/s or N⋅m⋅s for angular momentum versus kg⋅m/s or N⋅s for linear momentum. When calculating angular momentum as the product of the moment of inertia times the angular velocity, the angular velocity must be expressed in radians per second, where the radian assumes the dimensionless value of unity. (When performing dimensional analysis, it may be productive to use orientational analysis which treats radians as a base unit, but this is not done in the International system of units). The units of angular momentum can be interpreted as torque⋅time. An object with angular momentum of L N⋅m⋅s can be reduced to zero angular velocity by an angular impulse of L N⋅m⋅s.

The plane perpendicular to the axis of angular momentum and passing through the center of mass is sometimes called the invariable plane, because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered. One such plane is the invariable plane of the Solar System.

Angular momentum and torque

See also: Torque § Relationship with the angular momentum

Newton's second law of motion can be expressed mathematically, $${\displaystyle \mathbf{F}=m\mathbf{a},}$$ or force = mass × acceleration. The rotational equivalent for point particles may be derived as follows: $${\displaystyle \mathbf{L}=I\mathit{\omega }}$$ which means that the torque (i.e. the time derivative of the angular momentum) is $${\displaystyle \mathit{\tau }=\frac{dI}{dt}\mathit{\omega }+I\frac{d\mathit{\omega }}{dt}.}$$

Because the moment of inertia is ${\displaystyle m{r}^2}$, it follows that ${\displaystyle \frac{dI}{dt}=2mr\frac{dr}{dt}=2r{p}_{||}}$, and ${\displaystyle \frac{d\mathbf{L}}{dt}=I\frac{d\mathit{\omega }}{dt}+2r{p}_{||}\mathit{\omega },}$ which, reduces to $${\displaystyle \mathit{\tau }=I\mathit{\alpha }+2r{p}_{||}\mathit{\omega }.}$$ This is the rotational analog of Newton's second law. Note that the torque is not necessarily proportional or parallel to the angular acceleration (as one might expect). The reason for this is that the moment of inertia of a particle can change with time, something that cannot occur for ordinary mass.

Conservation of angular momentum

A figure skater in a spin uses conservation of angular momentum – decreasing her moment of inertia by drawing in her arms and legs increases her rotational speed.

General considerations

A rotational analog of Newton's third law of motion might be written, "In an isolated system, no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque about the same axis." Hence, angular momentum can be exchanged between objects in an isolated system, but total angular momentum before and after an exchange remains constant (is conserved).

Seen another way, a rotational analogue of Newton's first law of motion might be written, "A rigid body continues in a state of uniform rotation unless acted upon by an external influence." Thus with no external influence to act upon it, the original angular momentum of the system remains constant.

The conservation of angular momentum is used in analyzing central force motion. If the net force on some body is directed always toward some point, the center, then there is no torque on the body with respect to the center, as all of the force is directed along the radius vector, and none is perpendicular to the radius. Mathematically, torque ${\displaystyle \mathit{\tau }=\mathbf{r}\times \mathbf{F}=\mathbf{0},}$ because in this case ${\displaystyle \mathbf{r}}$ and ${\displaystyle \mathbf{F}}$ are parallel vectors. Therefore, the angular momentum of the body about the center is constant. This is the case with gravitational attraction in the orbits of planets and satellites, where the gravitational force is always directed toward the primary body and orbiting bodies conserve angular momentum by exchanging distance and velocity as they move about the primary. Central force motion is also used in the analysis of the Bohr model of the atom.

For a planet, angular momentum is distributed between the spin of the planet and its revolution in its orbit, and these are often exchanged by various mechanisms. The conservation of angular momentum in the Earth–Moon system results in the transfer of angular momentum from Earth to Moon, due to tidal torque the Moon exerts on the Earth. This in turn results in the slowing down of the rotation rate of Earth, at about 65.7 nanoseconds per day, and in gradual increase of the radius of Moon's orbit, at about 3.82 centimeters per year.

The torque caused by the two opposing forces F_g and −F_g causes a change in the angular momentum L in the direction of that torque (since torque is the time derivative of angular momentum). This causes the top to precess.

The conservation of angular momentum explains the angular acceleration of an ice skater as they bring their arms and legs close to the vertical axis of rotation. By bringing part of the mass of their body closer to the axis, they decrease their body's moment of inertia. Because angular momentum is the product of moment of inertia and angular velocity, if the angular momentum remains constant (is conserved), then the angular velocity (rotational speed) of the skater must increase.

The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron stars and black holes) when they are formed out of much larger and slower rotating stars.

Conservation is not always a full explanation for the dynamics of a system but is a key constraint. For example, a spinning top is subject to gravitational torque making it lean over and change the angular momentum about the nutation axis, but neglecting friction at the point of spinning contact, it has a conserved angular momentum about its spinning axis, and another about its precession axis. Also, in any planetary system, the planets, star(s), comets, and asteroids can all move in numerous complicated ways, but only so that the angular momentum of the system is conserved.

Noether's theorem states that every conservation law is associated with a symmetry (invariant) of the underlying physics. The symmetry associated with conservation of angular momentum is rotational invariance. The fact that the physics of a system is unchanged if it is rotated by any angle about an axis implies that angular momentum is conserved.

Relation to Newton's second law of motion

While angular momentum total conservation can be understood separately from Newton's laws of motion as stemming from Noether's theorem in systems symmetric under rotations, it can also be understood simply as an efficient method of calculation of results that can also be otherwise arrived at directly from Newton's second law, together with laws governing the forces of nature (such as Newton's third law, Maxwell's equations and Lorentz force). Indeed, given initial conditions of position and velocity for every point, and the forces at such a condition, one may use Newton's second law to calculate the second derivative of position, and solving for this gives full information on the development of the physical system with time. Note, however, that this is no longer true in quantum mechanics, due to the existence of particle spin, which is an angular momentum that cannot be described by the cumulative effect of point-like motions in space.

As an example, consider decreasing of the moment of inertia, e.g. when a figure skater is pulling in their hands, speeding up the circular motion. In terms of angular momentum conservation, we have, for angular momentum L, moment of inertia I and angular velocity ω: $${\displaystyle 0=dL=d(I\cdot \omega )=dI\cdot \omega +I\cdot d\omega }$$

Using this, we see that the change requires an energy of: $${\displaystyle dE=d\left(\frac{1}{2}I\cdot {\omega }^2\right)=\frac{1}{2}dI\cdot {\omega }^2+I\cdot \omega \cdot d\omega =-\frac{1}{2}dI\cdot {\omega }^2}$$ so that a decrease in the moment of inertia requires investing energy.

This can be compared to the work done as calculated using Newton's laws. Each point in the rotating body is accelerating, at each point of time, with radial acceleration of: $${\displaystyle -r\cdot {\omega }^2}$$

Let us observe a point of mass m, whose position vector relative to the center of motion is perpendicular to the z-axis at a given point of time, and is at a distance z. The centripetal force on this point, keeping the circular motion, is: $${\displaystyle -m\cdot z\cdot {\omega }^2}$$

Thus the work required for moving this point to a distance dz farther from the center of motion is: $${\displaystyle dW=-m\cdot z\cdot {\omega }^2\cdot dz=-m\cdot {\omega }^2\cdot d\left(\frac{1}{2}{z}^2\right)}$$

For a non-pointlike body one must integrate over this, with m replaced by the mass density per unit z. This gives: $${\displaystyle dW=-\frac{1}{2}dI\cdot {\omega }^2}$$ which is exactly the energy required for keeping the angular momentum conserved.

Note, that the above calculation can also be performed per mass, using kinematics only. Thus the phenomena of figure skater accelerating tangential velocity while pulling their hands in, can be understood as follows in layman's language: The skater's palms are not moving in a straight line, so they are constantly accelerating inwards, but do not gain additional speed because the accelerating is always done when their motion inwards is zero. However, this is different when pulling the palms closer to the body: The acceleration due to rotation now increases the speed; but because of the rotation, the increase in speed does not translate to a significant speed inwards, but to an increase of the rotation speed.

Stationary-action principle

In classical mechanics it can be shown that the rotational invariance of action functionals implies conservation of angular momentum. The action is defined in classical physics as a functional of positions, ${\displaystyle x_i(t)}$ often represented by the use of square brackets, and the final and initial times. It assumes the following form in cartesian coordinates:$${\displaystyle S\left([x_i];t_1,t_2\right)\equiv {\int }_{t_1}^{t_2}dt\left(\frac{1}{2}m\frac{d{x}_i}{dt}\frac{d{x}_i}{dt}-V(x_i)\right)}$$where the repeated indices indicate summation over the index. If the action is invariant of an infinitesimal transformation, it can be mathematically stated as: $\delta S=S\left([x_i+\delta x_i];t_1,t_2\right)-S\left([x_i];t_1,t_2\right)=0$.

Under the transformation, ${\displaystyle x_i\to x_i+\delta x_i}$, the action becomes: $${\displaystyle S\left([x_i+\delta x_i];t_1,t_2\right)={\int }_{t_1}^{t_2}dt\left(\frac{1}{2}m\frac{d(x_i+\delta x_i)}{dt}\frac{d(x_i+\delta x_i)}{dt}-V(x_i+\delta x_i)\right)}$$ where we can employ the expansion of the terms up-to first order in $\delta x_i$: $${\displaystyle \begin{array}{rl}\frac{d(x_i+\delta x_i)}{dt}\frac{d(x_i+\delta x_i)}{dt}& \simeq \frac{d{x}_i}{dt}\frac{d{x}_i}{dt}-2\frac{d^2{x}_i}{d{t}^2}\delta x_i+2\frac{d}{dt}\left(\delta x_i\frac{d{x}_i}{dt}\right)\\ V(x_i+\delta x_i)& \simeq V(x_i)+\delta x_i\frac{\mathrm{\partial }V}{\mathrm{\partial }{x}_i}\end{array}}$$giving the following change in action: $${\displaystyle S[x_i+\delta x_i]\simeq S[x_i]+{\int }_{t_1}^{t_2}dt\delta x_i\left(-\frac{\mathrm{\partial }V}{\mathrm{\partial }{x}_i}-m\frac{d^2{x}_i}{d{t}^2}\right)+m{\int }_{t_1}^{t_2}dt\frac{d}{dt}\left(\delta x_i\frac{d{x}_i}{dt}\right).}$$

Since all rotations can be expressed as matrix exponential of skew-symmetric matrices, i.e. as ${\displaystyle R(\hat{n},\theta )=e^{M\theta }}$ where ${\displaystyle M}$ is a skew-symmetric matrix and ${\displaystyle \theta }$ is angle of rotation, we can express the change of coordinates due to the rotation ${\displaystyle R(\hat{n},\delta \theta )}$, up-to first order of infinitesimal angle of rotation, ${\displaystyle \delta \theta }$ as: $${\displaystyle \delta x_i=M_{ij}{x}_j\delta \theta .}$$

Combining the equation of motion and rotational invariance of action, we get from the above equations that:$${\displaystyle 0=\delta S={\int }_{t_1}^{t_2}dt\frac{d}{dt}\left(m\frac{d{x}_i}{dt}\delta x_i\right)=M_{ij}\delta \theta m{x}_j\frac{d{x}_i}{dt}{|}_{t_1}^{t_2}}$$Since this is true for any matrix ${\displaystyle M_{ij}}$ that satisfies ${\displaystyle M_{ij}=-M_{ji},}$ it results in the conservation of the following quantity: $${\displaystyle {\ell }_{ij}(t):=m\left(x_i\frac{d{x}_j}{dt}-x_j\frac{d{x}_i}{dt}\right),}$$ as ${\displaystyle {\ell }_{ij}(t_1)={\ell }_{ij}(t_2)}$. This corresponds to the conservation of angular momentum throughout the motion.

Lagrangian formalism

In Lagrangian mechanics, angular momentum for rotation around a given axis, is the conjugate momentum of the generalized coordinate of the angle around the same axis. For example, ${\displaystyle L_z}$, the angular momentum around the z axis, is: $${\displaystyle L_z=\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{\dot{\theta }}_z}}$$ where ${\displaystyle \mathcal{L}}$ is the Lagrangian and ${\displaystyle {\theta }_z}$ is the angle around the z axis.

Note that ${\displaystyle {\dot{\theta }}_z}$, the time derivative of the angle, is the angular velocity ${\displaystyle {\omega }_z}$. Ordinarily, the Lagrangian depends on the angular velocity through the kinetic energy: The latter can be written by separating the velocity to its radial and tangential part, with the tangential part at the x-y plane, around the z-axis, being equal to: $${\displaystyle \sum _i\frac{1}{2}{m}_i{v_T}_i^2=\sum _i\frac{1}{2}{m}_i\left(x_i^2+y_i^2\right){{{\omega }_z}_i}^2}$$ where the subscript i stands for the i-th body, and ${\displaystyle m}$, ${\displaystyle v_T}$ and ${\displaystyle {\omega }_z}$stand for mass, tangential velocity around the z-axis and angular velocity around that axis, respectively.

For a body that is not point-like, with density ρ, we have instead: $${\displaystyle \frac{1}{2}\int \rho (x,y,z)\left(x_i^2+y_i^2\right){{{\omega }_z}_i}^{2}dxdy=\frac{1}{2}{I_z}_i{{{\omega }_z}_i}^2}$$ where integration runs over the area of the body, and I_z is the moment of inertia around the z-axis.

Thus, assuming the potential energy does not depend on ω_z (this assumption may fail for electromagnetic systems), we have the angular momentum of the ith object: $${\displaystyle \begin{array}{rl}{L_z}_i& =\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{{\omega }_z}_i}=\frac{\mathrm{\partial }{E}_k}{\mathrm{\partial }{{\omega }_z}_i}\\ & ={I_z}_i\cdot {{\omega }_z}_i\end{array}}$$

We have thus far rotated each object by a separate angle; we may also define an overall angle θ_z by which we rotate the whole system, thus rotating also each object around the z-axis, and have the overall angular momentum: $${\displaystyle L_z=\sum _i{I_z}_i\cdot {{\omega }_z}_i}$$

From Euler–Lagrange equations it then follows that: $${\displaystyle 0=\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{{\theta }_z}_i}-\frac{d}{dt}\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{{\dot{\theta }}_z}_i}\right)=\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{{\theta }_z}_i}-\frac{d{L_z}_i}{dt}}$$

Since the lagrangian is dependent upon the angles of the object only through the potential, we have: $${\displaystyle \frac{d{L_z}_i}{dt}=\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{{\theta }_z}_i}=-\frac{\mathrm{\partial }V}{\mathrm{\partial }{{\theta }_z}_i}}$$ which is the torque on the ith object.

Suppose the system is invariant to rotations, so that the potential is independent of an overall rotation by the angle θ_z (thus it may depend on the angles of objects only through their differences, in the form ${\displaystyle V({{\theta }_z}_i,{{\theta }_z}_j)=V({{\theta }_z}_i-{{\theta }_z}_j)}$). We therefore get for the total angular momentum: $${\displaystyle \frac{d{L}_z}{dt}=-\frac{\mathrm{\partial }V}{\mathrm{\partial }{\theta }_z}=0}$$ And thus the angular momentum around the z-axis is conserved.

This analysis can be repeated separately for each axis, giving conservation of the angular momentum vector. However, the angles around the three axes cannot be treated simultaneously as generalized coordinates, since they are not independent; in particular, two angles per point suffice to determine its position. While it is true that in the case of a rigid body, fully describing it requires, in addition to three translational degrees of freedom, also specification of three rotational degrees of freedom; however these cannot be defined as rotations around the Cartesian axes (see Euler angles). This caveat is reflected in quantum mechanics in the non-trivial commutation relations of the different components of the angular momentum operator.

Hamiltonian formalism

Equivalently, in Hamiltonian mechanics the Hamiltonian can be described as a function of the angular momentum. As before, the part of the kinetic energy related to rotation around the z-axis for the ith object is: $${\displaystyle \frac{1}{2}{I_z}_i{{{\omega }_z}_i}^2=\frac{{{L_z}_i}^2}{2{I_z}_i}}$$ which is analogous to the energy dependence upon momentum along the z-axis, ${\displaystyle \frac{{{p_z}_i}^2}{{2m}_i}}$.

Hamilton's equations relate the angle around the z-axis to its conjugate momentum, the angular momentum around the same axis: $${\displaystyle \begin{array}{rl}\frac{d{{\theta }_z}_i}{dt}& =\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }{L_z}_i}=\frac{{L_z}_i}{{I_z}_i}\\ \frac{d{L_z}_i}{dt}& =-\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }{{\theta }_z}_i}=-\frac{\mathrm{\partial }V}{\mathrm{\partial }{{\theta }_z}_i}\end{array}}$$

The first equation gives $${\displaystyle {L_z}_i={I_z}_i\cdot {{\dot{\theta }}_z}_i={I_z}_i\cdot {{\omega }_z}_i}$$

And so we get the same results as in the Lagrangian formalism.

Note, that for combining all axes together, we write the kinetic energy as: $${\displaystyle E_k=\frac{1}{2}\sum _i\frac{|{\mathbf{p}}_i{|}^2}{2m_i}=\sum _i\left(\frac{{p_r}_i^2}{2m_i}+\frac{1}{2}{{\mathbf{L}}_i}^{\mathsf{\text{T}}}{I_i}^{-1}{\mathbf{L}}_i\right)}$$ where p_r is the momentum in the radial direction, and the moment of inertia is a 3-dimensional matrix; bold letters stand for 3-dimensional vectors.

For point-like bodies we have: $${\displaystyle E_k=\sum _i\left(\frac{{p_r}_i^2}{2m_i}+\frac{|{\mathbf{L}}_i{|}^2}{2m_i{r_i}^2}\right)}$$

This form of the kinetic energy part of the Hamiltonian is useful in analyzing central potential problems, and is easily transformed to a quantum mechanical work frame (e.g. in the hydrogen atom problem).

Angular momentum in orbital mechanics

Main article: Specific angular momentum

While in classical mechanics the language of angular momentum can be replaced by Newton's laws of motion, it is particularly useful for motion in central potential such as planetary motion in the solar system. Thus, the orbit of a planet in the solar system is defined by its energy, angular momentum and angles of the orbit major axis relative to a coordinate frame.

In astrodynamics and celestial mechanics, a quantity closely related to angular momentum is defined as $${\displaystyle \mathbf{h}=\mathbf{r}\times \mathbf{v},}$$ called specific angular momentum. Note that ${\displaystyle \mathbf{L}=m\mathbf{h}.}$ Mass is often unimportant in orbital mechanics calculations, because motion of a body is determined by gravity. The primary body of the system is often so much larger than any bodies in motion about it that the gravitational effect of the smaller bodies on it can be neglected; it maintains, in effect, constant velocity. The motion of all bodies is affected by its gravity in the same way, regardless of mass, and therefore all move approximately the same way under the same conditions.

Solid bodies

Angular momentum is also an extremely useful concept for describing rotating rigid bodies such as a gyroscope or a rocky planet. For a continuous mass distribution with density function ρ(r), a differential volume element dV with position vector r within the mass has a mass element dm = ρ(r)dV. Therefore, the infinitesimal angular momentum of this element is: $${\displaystyle d\mathbf{L}=\mathbf{r}\times dm\mathbf{v}=\mathbf{r}\times \rho (\mathbf{r})dV\mathbf{v}=dV\mathbf{r}\times \rho (\mathbf{r})\mathbf{v}}$$ and integrating this differential over the volume of the entire mass gives its total angular momentum: $${\displaystyle \mathbf{L}={\int }_{V}dV\mathbf{r}\times \rho (\mathbf{r})\mathbf{v}}$$

In the derivation which follows, integrals similar to this can replace the sums for the case of continuous mass.

Collection of particles

The angular momentum of the particles i is the sum of the cross products R × MV + Σr_i × m_iv_i.

For a collection of particles in motion about an arbitrary origin, it is informative to develop the equation of angular momentum by resolving their motion into components about their own center of mass and about the origin. Given,

• ${\displaystyle m_i}$ is the mass of particle ${\displaystyle i}$,
• ${\displaystyle {\mathbf{R}}_i}$ is the position vector of particle ${\displaystyle i}$ w.r.t. the origin,
• ${\displaystyle {\mathbf{V}}_i}$ is the velocity of particle ${\displaystyle i}$ w.r.t. the origin,
• ${\displaystyle \mathbf{R}}$ is the position vector of the center of mass w.r.t. the origin,
• ${\displaystyle \mathbf{V}}$ is the velocity of the center of mass w.r.t. the origin,
• ${\displaystyle {\mathbf{r}}_i}$ is the position vector of particle ${\displaystyle i}$ w.r.t. the center of mass,
• ${\displaystyle {\mathbf{v}}_i}$ is the velocity of particle ${\displaystyle i}$ w.r.t. the center of mass,

The total mass of the particles is simply their sum, $${\displaystyle M=\sum _i{m}_i.}$$

The position vector of the center of mass is defined by, $${\displaystyle M\mathbf{R}=\sum _i{m}_i{\mathbf{R}}_i.}$$

By inspection,

${\displaystyle {\mathbf{R}}_i=\mathbf{R}+{\mathbf{r}}_i}$ and ${\displaystyle {\mathbf{V}}_i=\mathbf{V}+{\mathbf{v}}_i.}$

The total angular momentum of the collection of particles is the sum of the angular momentum of each particle,

${\displaystyle \mathbf{L}=\sum _i\left({\mathbf{R}}_i\times m_i{\mathbf{V}}_i\right)}$     (1)

Expanding ${\displaystyle {\mathbf{R}}_i}$,

${\displaystyle \begin{array}{rl}\mathbf{L}& =\sum _i\left[\left(\mathbf{R}+{\mathbf{r}}_i\right)\times m_i{\mathbf{V}}_i\right]\\ & =\sum _i\left[\mathbf{R}\times m_i{\mathbf{V}}_i+{\mathbf{r}}_i\times m_i{\mathbf{V}}_i\right]\end{array}}$

Expanding ${\displaystyle {\mathbf{V}}_i}$,

${\displaystyle \begin{array}{rl}\mathbf{L}& =\sum _i\left[\mathbf{R}\times m_i\left(\mathbf{V}+{\mathbf{v}}_i\right)+{\mathbf{r}}_i\times m_i(\mathbf{V}+{\mathbf{v}}_i)\right]\\ & =\sum _i\left[\mathbf{R}\times m_i\mathbf{V}+\mathbf{R}\times m_i{\mathbf{v}}_i+{\mathbf{r}}_i\times m_i\mathbf{V}+{\mathbf{r}}_i\times m_i{\mathbf{v}}_i\right]\\ & =\sum _i\mathbf{R}\times m_i\mathbf{V}+\sum _i\mathbf{R}\times m_i{\mathbf{v}}_i+\sum _i{\mathbf{r}}_i\times m_i\mathbf{V}+\sum _i{\mathbf{r}}_i\times m_i{\mathbf{v}}_i\end{array}}$

It can be shown that (see sidebar),

Prove that ${\displaystyle \sum _i{m}_i{\mathbf{r}}_i=\mathbf{0}}$ $${\displaystyle \begin{array}{rl}{\mathbf{r}}_i& ={\mathbf{R}}_i-\mathbf{R}\\ m_i{\mathbf{r}}_i& =m_i\left({\mathbf{R}}_i-\mathbf{R}\right)\\ \sum _i{m}_i{\mathbf{r}}_i& =\sum _i{m}_i\left({\mathbf{R}}_i-\mathbf{R}\right)\\ & =\sum _i(m_i{\mathbf{R}}_i-m_i\mathbf{R})\\ & =\sum _i{m}_i{\mathbf{R}}_i-\sum _i{m}_i\mathbf{R}\\ & =\sum _i{m}_i{\mathbf{R}}_i-\left(\sum _i{m}_i\right)\mathbf{R}\\ & =\sum _i{m}_i{\mathbf{R}}_i-M\mathbf{R}\end{array}}$$ which, by the definition of the center of mass, is ${\displaystyle \mathbf{0},}$ and similarly for $\sum _i{m}_i{\mathbf{v}}_i.$

${\displaystyle \sum _i{m}_i{\mathbf{r}}_i=\mathbf{0}}$ and ${\displaystyle \sum _i{m}_i{\mathbf{v}}_i=\mathbf{0},}$

therefore the second and third terms vanish,

${\displaystyle \mathbf{L}=\sum _i\mathbf{R}\times m_i\mathbf{V}+\sum _i{\mathbf{r}}_i\times m_i{\mathbf{v}}_i.}$

The first term can be rearranged,

${\displaystyle \sum _i\mathbf{R}\times m_i\mathbf{V}=\mathbf{R}\times \sum _i{m}_i\mathbf{V}=\mathbf{R}\times M\mathbf{V},}$

and total angular momentum for the collection of particles is finally,

${\displaystyle \mathbf{L}=\mathbf{R}\times M\mathbf{V}+\sum _i{\mathbf{r}}_i\times m_i{\mathbf{v}}_i}$     (2)

The first term is the angular momentum of the center of mass relative to the origin. Similar to § Single particle, below, it is the angular momentum of one particle of mass M at the center of mass moving with velocity V. The second term is the angular momentum of the particles moving relative to the center of mass, similar to § Fixed center of mass, below. The result is general—the motion of the particles is not restricted to rotation or revolution about the origin or center of mass. The particles need not be individual masses, but can be elements of a continuous distribution, such as a solid body.

Rearranging equation (2) by vector identities, multiplying both terms by "one", and grouping appropriately, $${\displaystyle \begin{array}{rl}\mathbf{L}& =M(\mathbf{R}\times \mathbf{V})+\sum _i\left[m_i\left({\mathbf{r}}_i\times {\mathbf{v}}_i\right)\right],\\ & =\frac{R^2}{R^2}M\left(\mathbf{R}\times \mathbf{V}\right)+\sum _i\left[\frac{r_i^2}{r_i^2}{m}_i\left({\mathbf{r}}_i\times {\mathbf{v}}_i\right)\right],\\ & =R^{2}M\left(\frac{\mathbf{R}\times \mathbf{V}}{R^2}\right)+\sum _i\left[r_i^2{m}_i\left(\frac{{\mathbf{r}}_i\times {\mathbf{v}}_i}{r_i^2}\right)\right],\end{array}}$$ gives the total angular momentum of the system of particles in terms of moment of inertia ${\displaystyle I}$ and angular velocity ${\displaystyle \mathit{\omega }}$,

${\displaystyle \mathbf{L}=I_R{\mathit{\omega }}_R+\sum _i{I}_i{\mathit{\omega }}_i.}$     (3)

Single particle case

In the case of a single particle moving about the arbitrary origin, $${\displaystyle \begin{array}{rl}{\mathbf{r}}_i& ={\mathbf{v}}_i=\mathbf{0},\\ \mathbf{r}& =\mathbf{R},\\ \mathbf{v}& =\mathbf{V},\\ m& =M,\end{array}}$$ $${\displaystyle \sum _i{\mathbf{r}}_i\times m_i{\mathbf{v}}_i=\mathbf{0},}$$ $${\displaystyle \sum _i{I}_i{\mathit{\omega }}_i=\mathbf{0},}$$ and equations (2) and (3) for total angular momentum reduce to, $${\displaystyle \mathbf{L}=\mathbf{R}\times m\mathbf{V}=I_R{\mathit{\omega }}_R.}$$

Case of a fixed center of mass

For the case of the center of mass fixed in space with respect to the origin, $${\displaystyle \mathbf{V}=\mathbf{0},}$$ $${\displaystyle \mathbf{R}\times M\mathbf{V}=\mathbf{0},}$$ $${\displaystyle I_R{\mathit{\omega }}_R=\mathbf{0},}$$ and equations (2) and (3) for total angular momentum reduce to, $${\displaystyle \mathbf{L}=\sum _i{\mathbf{r}}_i\times m_i{\mathbf{v}}_i=\sum _i{I}_i{\mathit{\omega }}_i.}$$

Angular momentum in general relativity

The 3-angular momentum as a bivector (plane element) and axial vector, of a particle of mass m with instantaneous 3-position x and 3-momentum p.

Main article: Relativistic angular momentum

In modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum – see below) is described using a different formalism, instead of a classical pseudovector. In this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance. As a result, angular momentum is generally not conserved locally for general curved spacetimes, unless they have rotational symmetry; whereas globally the notion of angular momentum itself only makes sense if the spacetime is asymptotically flat. If the spacetime is only axially symmetric like for the Kerr metric, the total angular momentum is not conserved but ${\displaystyle p_{\varphi }}$ is conserved which is related to the invariance of rotating around the symmetry-axis, where note that ${\displaystyle p_{\varphi }=g_{\varphi \mu }{p}^{\mu }=m{g}_{\mu \varphi }d{X}^{\mu }/d\tau }$ where ${\displaystyle g_{\mu \nu }}$ is the metric, ${\displaystyle m=\sqrt{|p_{\mu }{p}^{\mu }|}}$ is the rest mass, ${\displaystyle d{X}^{\mu }/d\tau }$ is the four-velocity, and ${\displaystyle X^{\mu }=(t,r,\theta ,\varphi )}$ is the four-position in spherical coordinates.

In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element: $${\displaystyle \mathbf{L}=\mathbf{r}\wedge \mathbf{p},}$$ in which the exterior product (∧) replaces the cross product (×) (these products have similar characteristics but are nonequivalent). This has the advantage of a clearer geometric interpretation as a plane element, defined using the vectors x and p, and the expression is true in any number of dimensions. In Cartesian coordinates: $${\displaystyle \begin{array}{rl}\mathbf{L}& =\left(x{p}_y-y{p}_x\right){\mathbf{e}}_x\wedge {\mathbf{e}}_y+\left(y{p}_z-z{p}_y\right){\mathbf{e}}_y\wedge {\mathbf{e}}_z+\left(z{p}_x-x{p}_z\right){\mathbf{e}}_z\wedge {\mathbf{e}}_x\\ & =L_{xy}{\mathbf{e}}_x\wedge {\mathbf{e}}_y+L_{yz}{\mathbf{e}}_y\wedge {\mathbf{e}}_z+L_{zx}{\mathbf{e}}_z\wedge {\mathbf{e}}_x,\end{array}}$$ or more compactly in index notation: $${\displaystyle L_{ij}=x_i{p}_j-x_j{p}_i.}$$

The angular velocity can also be defined as an anti-symmetric second order tensor, with components ω_ij. The relation between the two anti-symmetric tensors is given by the moment of inertia which must now be a fourth order tensor: $${\displaystyle L_{ij}=I_{ijk\ell }{\omega }_{k\ell }.}$$

Again, this equation in L and ω as tensors is true in any number of dimensions. This equation also appears in the geometric algebra formalism, in which L and ω are bivectors, and the moment of inertia is a mapping between them.

In relativistic mechanics, the relativistic angular momentum of a particle is expressed as an anti-symmetric tensor of second order: $${\displaystyle M_{\alpha \beta }=X_{\alpha }{P}_{\beta }-X_{\beta }{P}_{\alpha }}$$ in terms of four-vectors, namely the four-position X and the four-momentum P, and absorbs the above L together with the moment of mass, i.e., the product of the relativistic mass of the particle and its center of mass, which can be thought of as describing the motion of its center of mass, since mass–energy is conserved.

In each of the above cases, for a system of particles the total angular momentum is just the sum of the individual particle angular momenta, and the center of mass is for the system.

Angular momentum in quantum mechanics

Main article: Angular momentum operator

In quantum mechanics, angular momentum (like other quantities) is expressed as an operator, and its one-dimensional projections have quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, implying that at any time, only one projection (also called "component") can be measured with definite precision; the other two then remain uncertain. Because of this, the axis of rotation of a quantum particle is undefined. Quantum particles do possess a type of non-orbital angular momentum called "spin", but this angular momentum does not correspond to a spinning motion. In relativistic quantum mechanics the above relativistic definition becomes a tensorial operator.

Spin, orbital, and total angular momentum

Main article: Spin (physics)

Angular momenta of a classical object.

• Left: "spin" angular momentum S is really orbital angular momentum of the object at every point.
• Right: extrinsic orbital angular momentum L about an axis.
• Top: the moment of inertia tensor I and angular velocity ω (L is not always parallel to ω).^[36]
• Bottom: momentum p and its radial position r from the axis. The total angular momentum (spin plus orbital) is J. For a quantum particle the interpretations are different; particle spin does not have the above interpretation.

The classical definition of angular momentum as ${\displaystyle \mathbf{L}=\mathbf{r}\times \mathbf{p}}$ can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator. L is then an operator, specifically called the orbital angular momentum operator. The components of the angular momentum operator satisfy the commutation relations of the Lie algebra SO(3). Indeed, these operators are precisely the infinitesimal action of the rotation group on the quantum Hilbert space. (See also the discussion below of the angular momentum operators as the generators of rotations.)

However, in quantum physics, there is another type of angular momentum, called spin angular momentum, represented by the spin operator S. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: spin is an intrinsic property of a particle, unrelated to any sort of motion in space and fundamentally different from orbital angular momentum. All elementary particles have a characteristic spin, which is nonzero for all elementary particles other than the Higgs boson (which has spin 0, making it the only known elementary scalar boson). For example, electrons have "spin 1/2" (this actually means "spin ħ/2"), photons have "spin 1" (this actually means "spin ħ"), and pi-mesons have spin 0.

Finally, there is total angular momentum J, which combines both the spin and orbital angular momentum of all particles and fields. (For one particle, J = L + S.) Conservation of angular momentum applies to J, but not to L or S; for example, the spin–orbit interaction allows angular momentum to transfer back and forth between L and S, with the total remaining constant. Electrons and photons need not have integer-based values for total angular momentum, but can also have half-integer values.

In molecules the total angular momentum F is the sum of the rovibronic (orbital) angular momentum N, the electron spin angular momentum S, and the nuclear spin angular momentum I. For electronic singlet states the rovibronic angular momentum is denoted J rather than N. As explained by Van Vleck, the components of the molecular rovibronic angular momentum referred to molecule-fixed axes have different commutation relations from those for the components about space-fixed axes.

Quantization

Further information: Angular momentum operator

In quantum mechanics, angular momentum is quantized – that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. For any system, the following restrictions on measurement results apply, where ${\displaystyle \hslash }$ is the reduced Planck constant and ${\displaystyle \hat{n}}$ is any Euclidean vector such as x, y, or z:

If you measure...	The result can be...
${\displaystyle L_{\hat{n}}}$	${\displaystyle \dots ,-2\hslash ,-\hslash ,0,\hslash ,2\hslash ,\dots }$
${\displaystyle S_{\hat{n}}}$ or ${\displaystyle J_{\hat{n}}}$	${\displaystyle \dots ,-\frac{3}{2}\hslash ,-\hslash ,-\frac{1}{2}\hslash ,0,\frac{1}{2}\hslash ,\hslash ,\frac{3}{2}\hslash ,\dots }$
${\displaystyle \begin{array}{rl}& L^2\\ =& L_x^2+L_y^2+L_z^2\end{array}}$	${\displaystyle \left[{\hslash }^{2}n(n+1)\right]}$, where ${\displaystyle n=0,1,2,\dots }$
${\displaystyle S^2}$ or ${\displaystyle J^2}$	${\displaystyle \left[{\hslash }^{2}n(n+1)\right]}$, where ${\displaystyle n=0,\frac{1}{2},1,\frac{3}{2},\dots }$

In this standing wave on a circular string, the circle is broken into exactly 8 wavelengths. A standing wave like this can have 0, 1, 2, or any integer number of wavelengths around the circle, but it cannot have a non-integer number of wavelengths like 8.3. In quantum mechanics, angular momentum is quantized for a similar reason.

The reduced Planck constant ${\displaystyle \hslash }$ is tiny by everyday standards, about 10⁻³⁴ J s, and therefore this quantization does not noticeably affect the angular momentum of macroscopic objects. However, it is very important in the microscopic world. For example, the structure of electron shells and subshells in chemistry is significantly affected by the quantization of angular momentum.

Quantization of angular momentum was first postulated by Niels Bohr in his model of the atom and was later predicted by Erwin Schrödinger in his Schrödinger equation.

Uncertainty

In the definition ${\displaystyle \mathbf{L}=\mathbf{r}\times \mathbf{p}}$, six operators are involved: The position operators ${\displaystyle r_x}$, ${\displaystyle r_y}$, ${\displaystyle r_z}$, and the momentum operators ${\displaystyle p_x}$, ${\displaystyle p_y}$, ${\displaystyle p_z}$. However, the Heisenberg uncertainty principle tells us that it is not possible for all six of these quantities to be known simultaneously with arbitrary precision. Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one axis.

The uncertainty is closely related to the fact that different components of an angular momentum operator do not commute, for example ${\displaystyle L_x{L}_y\ne L_y{L}_x}$. (For the precise commutation relations, see angular momentum operator.)

Total angular momentum as generator of rotations

As mentioned above, orbital angular momentum L is defined as in classical mechanics: ${\displaystyle \mathbf{L}=\mathbf{r}\times \mathbf{p}}$, but total angular momentum J is defined in a different, more basic way: J is defined as the "generator of rotations". More specifically, J is defined so that the operator $${\displaystyle R(\hat{n},\varphi )\equiv \exp \left(-\frac{i}{\hslash }\varphi \mathbf{J}\cdot \hat{\mathbf{n}}\right)}$$ is the rotation operator that takes any system and rotates it by angle ${\displaystyle \varphi }$ about the axis ${\displaystyle \hat{\mathbf{n}}}$. (The "exp" in the formula refers to operator exponential.) To put this the other way around, whatever our quantum Hilbert space is, we expect that the rotation group SO(3) will act on it. There is then an associated action of the Lie algebra so(3) of SO(3); the operators describing the action of so(3) on our Hilbert space are the (total) angular momentum operators.

The relationship between the angular momentum operator and the rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics. The close relationship between angular momentum and rotations is reflected in Noether's theorem that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant.

Angular momentum in electrodynamics

See also: Momentum (Particle in field)

When describing the motion of a charged particle in an electromagnetic field, the canonical momentum P (derived from the Lagrangian for this system) is not gauge invariant. As a consequence, the canonical angular momentum L = r × P is not gauge invariant either. Instead, the momentum that is physical, the so-called kinetic momentum (used throughout this article), is (in SI units) $${\displaystyle \mathbf{p}=m\mathbf{v}=\mathbf{P}-e\mathbf{A}}$$ where e is the electric charge of the particle and A the magnetic vector potential of the electromagnetic field. The gauge-invariant angular momentum, that is kinetic angular momentum, is given by $${\displaystyle \mathbf{K}=\mathbf{r}\times (\mathbf{P}-e\mathbf{A})}$$

The interplay with quantum mechanics is discussed further in the article on canonical commutation relations.

Angular momentum in optics

In classical Maxwell electrodynamics the Poynting vector is a linear momentum density of electromagnetic field. $${\displaystyle \mathbf{S}(\mathbf{r},t)={ϵ}_0{c}^2\mathbf{E}(\mathbf{r},t)\times \mathbf{B}(\mathbf{r},t).}$$

The angular momentum density vector ${\displaystyle \mathbf{L}(\mathbf{r},t)}$ is given by a vector product as in classical mechanics: $${\displaystyle \mathbf{L}(\mathbf{r},t)={ϵ}_0{\mu }_0\mathbf{r}\times \mathbf{S}(\mathbf{r},t).}$$

The above identities are valid locally, i.e. in each space point ${\displaystyle \mathbf{r}}$ in a given moment ${\displaystyle t}$.

Angular momentum in nature and the cosmos

Tropical cyclones and other related weather phenomena involve conservation of angular momentum in order to explain the dynamics. Winds revolve slowly around low pressure systems, mainly due to the coriolis effect. If the low pressure intensifies and the slowly circulating air is drawn toward the center, the molecules must speed up in order to conserve angular momentum. By the time they reach the center, the speeds become destructive.

Johannes Kepler determined the laws of planetary motion without knowledge of conservation of momentum. However, not long after his discovery their derivation was determined from conservation of angular momentum. Planets move more slowly the further they are out in their elliptical orbits, which is explained intuitively by the fact that orbital angular momentum is proportional to the radius of the orbit. Since the mass does not change and the angular momentum is conserved, the velocity drops.

Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite (e.g. the Moon) and the primary planet that it orbits (e.g. Earth). The gravitational torque between the Moon and the tidal bulge of Earth causes the Moon to be constantly promoted to a slightly higher orbit (~3.8 cm per year) and Earth to be decelerated (by −25.858 ± 0.003″/cy²) in its rotation (the length of the day increases by ~1.7 ms per century, +2.3 ms from tidal effect and −0.6 ms from post-glacial rebound). The Earth loses angular momentum which is transferred to the Moon such that the overall angular momentum is conserved.

Angular momentum in engineering and technology

Examples of using conservation of angular momentum for practical advantage are abundant. In engines such as steam engines or internal combustion engines, a flywheel is needed to efficiently convert the lateral motion of the pistons to rotational motion.

Inertial navigation systems explicitly use the fact that angular momentum is conserved with respect to the inertial frame of space. Inertial navigation is what enables submarine trips under the polar ice cap, but are also crucial to all forms of modern navigation.

Rifled bullets use the stability provided by conservation of angular momentum to be more true in their trajectory. The invention of rifled firearms and cannons gave their users significant strategic advantage in battle, and thus were a technological turning point in history.

History

Isaac Newton, in the Principia, hinted at angular momentum in his examples of the first law of motion,

A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.

He did not further investigate angular momentum directly in the Principia, saying:

From such kind of reflexions also sometimes arise the circular motions of bodies about their own centers. But these are cases which I do not consider in what follows; and it would be too tedious to demonstrate every particular that relates to this subject.

However, his geometric proof of the law of areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

Law of Areas

Main articles: Classical central-force problem and Areal velocity

Newton's derivation

Newton's derivation of the area law using geometric means

As a planet orbits the Sun, the line between the Sun and the planet sweeps out equal areas in equal intervals of time. This had been known since Kepler expounded his second law of planetary motion. Newton derived a unique geometric proof, and went on to show that the attractive force of the Sun's gravity was the cause of all of Kepler's laws.

During the first interval of time, an object is in motion from point A to point B. Undisturbed, it would continue to point c during the second interval. When the object arrives at B, it receives an impulse directed toward point S. The impulse gives it a small added velocity toward S, such that if this were its only velocity, it would move from B to V during the second interval. By the rules of velocity composition, these two velocities add, and point C is found by construction of parallelogram BcCV. Thus the object's path is deflected by the impulse so that it arrives at point C at the end of the second interval. Because the triangles SBc and SBC have the same base SB and the same height Bc or VC, they have the same area. By symmetry, triangle SBc also has the same area as triangle SAB, therefore the object has swept out equal areas SAB and SBC in equal times.

At point C, the object receives another impulse toward S, again deflecting its path during the third interval from d to D. Thus it continues to E and beyond, the triangles SAB, SBc, SBC, SCd, SCD, SDe, SDE all having the same area. Allowing the time intervals to become ever smaller, the path ABCDE approaches indefinitely close to a continuous curve.

Note that because this derivation is geometric, and no specific force is applied, it proves a more general law than Kepler's second law of planetary motion. It shows that the Law of Areas applies to any central force, attractive or repulsive, continuous or non-continuous, or zero.

Conservation of angular momentum in the law of areas

The proportionality of angular momentum to the area swept out by a moving object can be understood by realizing that the bases of the triangles, that is, the lines from S to the object, are equivalent to the radius r, and that the heights of the triangles are proportional to the perpendicular component of velocity v_⊥. Hence, if the area swept per unit time is constant, then by the triangular area formula ⁠1/2⁠(base)(height), the product (base)(height) and therefore the product rv_⊥ are constant: if r and the base length are decreased, v_⊥ and height must increase proportionally. Mass is constant, therefore angular momentum rmv_⊥ is conserved by this exchange of distance and velocity.

In the case of triangle SBC, area is equal to ⁠1/2⁠(SB)(VC). Wherever C is eventually located due to the impulse applied at B, the product (SB)(VC), and therefore rmv_⊥ remain constant. Similarly so for each of the triangles.

Another areal proof of conservation of angular momentum for any central force uses Mamikon's sweeping tangents theorem.

After Newton

Leonhard Euler, Daniel Bernoulli, and Patrick d'Arcy all understood angular momentum in terms of conservation of areal velocity, a result of their analysis of Kepler's second law of planetary motion. It is unlikely that they realized the implications for ordinary rotating matter.

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them.

Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.

In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation—his invariable plane.

Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".

In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.

William J. M. Rankine's 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:

... a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation.

In an 1872 edition of the same book, Rankine stated that "The term angular momentum was introduced by Mr. Hayward," probably referring to R.B. Hayward's article On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space with Applications, which was introduced in 1856, and published in 1864. Rankine was mistaken, as numerous publications feature the term starting in the late 18th to early 19th centuries. However, Hayward's article apparently was the first use of the term and the concept seen by much of the English-speaking world. Before this, angular momentum was typically referred to as "momentum of rotation" in English.

Evolutionary linguistics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Evolutionary_linguistics

Evolutionary linguistics or Darwinian linguistics is a sociobiological approach to the study of language. Evolutionary linguists consider linguistics as a subfield of sociobiology and evolutionary psychology. The approach is also closely linked with evolutionary anthropology, cognitive linguistics and biolinguistics. Studying languages as the products of nature, it is interested in the biological origin and development of language. Evolutionary linguistics is contrasted with humanistic approaches, especially structural linguistics.

A main challenge in this research is the lack of empirical data: there are no archaeological traces of early human language. Computational biological modelling and clinical research with artificial languages have been employed to fill in gaps of knowledge. Although biology is understood to shape the brain, which processes language, there is no clear link between biology and specific human language structures or linguistic universals.

For lack of a breakthrough in the field, there have been numerous debates about what kind of natural phenomenon language might be. Some researchers focus on the innate aspects of language. It is suggested that grammar has emerged adaptationally from the human genome, bringing about a language instinct; or that it depends on a single mutation which has caused a language organ to appear in the human brain. This is hypothesized to result in a crystalline grammatical structure underlying all human languages. Others suggest language is not crystallized, but fluid and ever-changing. Others, yet, liken languages to living organisms. Languages are considered analogous to a parasite or populations of mind-viruses. There is so far little scientific evidence for any of these claims, and some of them have been labelled as pseudoscience.

History

1863–1945: social Darwinism

Although pre-Darwinian theorists had compared languages to living organisms as a metaphor, the comparison was first taken literally in 1863 by the historical linguist August Schleicher who was inspired by Charles Darwin's On the Origin of Species. At the time there was not enough evidence to prove that Darwin's theory of natural selection was correct. Schleicher proposed that linguistics could be used as a testing ground for the study of the evolution of species. A review of Schleicher's book Darwinism as Tested by the Science of Language appeared in the first issue of Nature journal in 1870. Darwin reiterated Schleicher's proposition in his 1871 book The Descent of Man, claiming that languages are comparable to species, and that language change occurs through natural selection as words 'struggle for life'. Darwin believed that languages had evolved from animal mating calls. Darwinists considered the concept of language creation as unscientific.

August Schleicher and his friend Ernst Haeckel were keen gardeners and regarded the study of cultures as a type of botany, with different species competing for the same living space. Similar ideas became later advocated by politicians who wanted to appeal to working class voters, not least by the national socialists who subsequently included the concept of struggle for living space in their agenda. Highly influential until the end of World War II, social Darwinism was eventually banished from human sciences, leading to a strict separation of natural and sociocultural studies.

This gave rise to the dominance of structural linguistics in Europe. There had long been a dispute between the Darwinists and the French intellectuals with the topic of language evolution famously having been banned by the Paris Linguistic Society as early as in 1866. Ferdinand de Saussure proposed structuralism to replace evolutionary linguistics in his Course in General Linguistics, published posthumously in 1916. The structuralists rose to academic political power in human and social sciences in the aftermath of the student revolts of Spring 1968, establishing the Sorbonne as an international centrepoint of humanistic thinking.

From 1959 onwards: genetic determinism

In the United States, structuralism was however fended off by the advocates of behavioural psychology; a linguistics framework nicknamed as 'American structuralism'. It was eventually replaced by the approach of Noam Chomsky who published a modification of Louis Hjelmslev's formal structuralist theory, claiming that syntactic structures are innate. An active figure in peace demonstrations in the 1950s and 1960s, Chomsky rose to academic political power following Spring 1968 at the MIT.

Chomsky became an influential opponent of the French intellectuals during the following decades, and his supporters successfully confronted the post-structuralists in the Science Wars of the late 1990s. The shift of the century saw a new academic funding policy where interdisciplinary research became favoured, effectively directing research funds to biological humanities. The decline of structuralism was evident by 2015 with Sorbonne having lost its former spirit.

Chomsky eventually claimed that syntactic structures are caused by a random mutation in the human genome, proposing a similar explanation for other human faculties such as ethics. But Steven Pinker argued in 1990 that they are the outcome of evolutionary adaptations.

From 1976 onwards: Neo-Darwinism

At the same time when the Chomskyan paradigm of biological determinism defeated humanism, it was losing its own clout within sociobiology. It was reported likewise in 2015 that generative grammar was under fire in applied linguistics and in the process of being replaced with usage-based linguistics; a derivative of Richard Dawkins's memetics. It is a concept of linguistic units as replicators. Following the publication of memetics in Dawkins's 1976 nonfiction bestseller The Selfish Gene, many biologically inclined linguists, frustrated with the lack of evidence for Chomsky's Universal Grammar, grouped under different brands including a framework called Cognitive Linguistics (with capitalised initials), and 'functional' (adaptational) linguistics (not to be confused with functional linguistics) to confront both Chomsky and the humanists. The replicator approach is today dominant in evolutionary linguistics, applied linguistics, cognitive linguistics and linguistic typology; while the generative approach has maintained its position in general linguistics, especially syntax; and in computational linguistics.

View of linguistics

Evolutionary linguistics is part of a wider framework of Universal Darwinism. In this view, linguistics is seen as an ecological environment for research traditions struggling for the same resources. According to David Hull, these traditions correspond to species in biology. Relationships between research traditions can be symbiotic, competitive or parasitic. An adaptation of Hull's theory in linguistics is proposed by William Croft. He argues that the Darwinian method is more advantageous than linguistic models based on physics, structuralist sociology, or hermeneutics.

Approaches

Evolutionary linguistics is often divided into functionalism and formalism, concepts which are not to be confused with functionalism and formalism in the humanistic reference. Functional evolutionary linguistics considers languages as adaptations to human mind. The formalist view regards them as crystallised or non-adaptational.

Functionalism (adaptationism)

The adaptational view of language is advocated by various frameworks of cognitive and evolutionary linguistics, with the terms 'functionalism' and 'Cognitive Linguistics' often being equated. It is hypothesised that the evolution of the animal brain provides humans with a mechanism of abstract reasoning which is a 'metaphorical' version of image-based reasoning. Language is not considered as a separate area of cognition, but as coinciding with general cognitive capacities, such as perception, attention, motor skills, and spatial and visual processing. It is argued to function according to the same principles as these.

It is thought that the brain links action schemes to form–meaning pairs which are called constructions. Cognitive linguistic approaches to syntax are called cognitive and construction grammar. Also deriving from memetics and other cultural replicator theories, these can study the natural or social selection and adaptation of linguistic units. Adaptational models reject a formal systemic view of language and consider language as a population of linguistic units.

The bad reputation of social Darwinism and memetics has been discussed in the literature, and recommendations for new terminology have been given. What correspond to replicators or mind-viruses in memetics are called linguemes in Croft's theory of Utterance Selection (TUS), and likewise linguemes or constructions in construction grammar and usage-based linguistics; and metaphors, frames or schemas in cognitive and construction grammar. The reference of memetics has been largely replaced with that of a Complex Adaptive System. In current linguistics, this term covers a wide range of evolutionary notions while maintaining the Neo-Darwinian concepts of replication and replicator population.

Functional evolutionary linguistics is not to be confused with functional humanistic linguistics.

Formalism (structuralism)

Advocates of formal evolutionary explanation in linguistics argue that linguistic structures are crystallised. Inspired by 19th century advances in crystallography, Schleicher argued that different types of languages are like plants, animals and crystals. The idea of linguistic structures as frozen drops was revived in tagmemics, an approach to linguistics with the goal to uncover divine symmetries underlying all languages, as if caused by the Creation.

In modern biolinguistics, the X-bar tree is argued to be like natural systems such as ferromagnetic droplets and botanic forms. Generative grammar considers syntactic structures similar to snowflakes. It is hypothesised that such patterns are caused by a mutation in humans.

The formal–structural evolutionary aspect of linguistics is not to be confused with structural linguistics.

Evidence

There was some hope of a breakthrough with the discovery of the FOXP2 gene. There is little support, however, for the idea that FOXP2 is 'the grammar gene' or that it had much to do with the relatively recent emergence of syntactical speech. The idea that people have a language instinct is disputed. Memetics is sometimes discredited as pseudoscience and neurological claims made by evolutionary cognitive linguists have been likened to pseudoscience. All in all, there does not appear to be any evidence for the basic tenets of evolutionary linguistics beyond the fact that language is processed by the brain, and brain structures are shaped by genes.

Criticism

Evolutionary linguistics has been criticised by advocates of (humanistic) structural and functional linguistics. Ferdinand de Saussure commented on 19th century evolutionary linguistics:

"Language was considered a specific sphere, a fourth natural kingdom; this led to methods of reasoning which would have caused astonishment in other sciences. Today one cannot read a dozen lines written at that time without being struck by absurdities of reasoning and by the terminology used to justify these absurdities"

Mark Aronoff, however, argues that historical linguistics had its golden age during the time of Schleicher and his supporters, enjoying a place among the hard sciences, and considers the return of Darwinian linguistics as a positive development. Esa Itkonen nonetheless deems the revival of Darwinism as a hopeless enterprise:

"There is ... an application of intelligence in linguistic change which is absent in biological evolution; and this suffices to make the two domains totally disanalogous ... [Grammaticalisation depends on] cognitive processes, ultimately serving the goal of problem solving, which intelligent entities like humans must perform all the time, but which biological entities like genes cannot perform. Trying to eliminate this basic difference leads to confusion."

Itkonen also points out that the principles of natural selection are not applicable because language innovation and acceptance have the same source which is the speech community. In biological evolution, mutation and selection have different sources. This makes it possible for people to change their languages, but not their genotype.

Inheritance

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Inheritance

From William Hogarth's A Rake's Progress. "The Young Heir Takes Possession Of The Miser's Effects".

Inheritance is the practice of receiving private property, titles, debts, entitlements, privileges, rights, and obligations upon the death of an individual. The rules of inheritance differ among societies and have changed over time. In legal terms, succession is the process by which a deceased person's rights and property are transferred to their heirs, while inheritance is the property or assets those heirs receive.

Succession may occur either under the generally applicable statutory rules, referred to as intestate succession, or in accordance with the provisions outlined in a valid will. A will often must be attested by a notary or by other lawful means to be valid.

Legal systems can differ significantly in how property passes from a deceased person to their heirs, with common law jurisdictions typically requiring formal probate procedures, while civil law systems often allow heirs to acquire ownership automatically by operation of law - the principle of saisine or seizin (Quebec).

Terminology

In law, an heir (fem (obs): heiress) is a person who is entitled to receive a share of property from a decedent (a person who died), subject to the rules of inheritance in the jurisdiction where the decedent was a citizen, or where the decedent died or owned property at the time of death.

The inheritance may be either under the terms of a will or by intestacy laws if the deceased had no will. However, the will must comply with the laws of the jurisdiction at the time it was created, or it will be declared invalid - for example, some states do not recognise handwritten wills as valid, or only in specific circumstances - and the intestacy laws then apply.

The exclusion from inheritance of a person who was an heir in a previous will, or would be expected to inherit under the laws of intestate succession, is termed disinheritance.

A person does not become an heir before the death of the deceased, since the exact identity of the persons entitled to inherit is determined only then. Members of ruling noble or royal houses who are expected to become heirs are called heirs apparent if first in line and incapable of being displaced from inheriting by another claim; otherwise, they are heirs presumptive. There is a further concept of joint inheritance, pending renunciation by all but one, which is called coparceny.

In modern law, the terms ‘'inheritance'’ and '‘heir’' apply only to property passed by intestate succession – that is, from a person who dies without a will. Property distributed under a will passes to beneficiaries, who may be called devisees for real property, legatees for money, and recipients of bequests for other personal property.

Except in some jurisdictions where a person cannot be legally disinherited (such as the US state of Louisiana, which allows disinheritance only under specifically enumerated circumstances), a person who would otherwise be an heir may be disinherited completely under the terms of a will (an example is that of the will of comedian Jerry Lewis; his will specifically disinherited his six children by his first wife, and their descendants, leaving his entire estate to his second wife).

Inheritance has been compared to nepotism.

History

Further information: Historical inheritance systems

Detailed anthropological and sociological studies have been conducted on customs of patrimonial inheritance, in which only male children can inherit. Some cultures also employ matrilineal succession, where property can only pass along the female line, most commonly to the decedent's sister's sons, but also, in some societies, to the mother and her daughters. Some ancient societies and most modern states employ egalitarian inheritance, without discrimination based on gender and/or birth order.

Religious laws about inheritance

Jewish laws

The inheritance is patrimonial. The father —that is, the owner of the land— bequeaths only to his male descendants. According to the Law of Moses, the firstborn son was entitled to receive twice as much of his father's inheritance as the other sons (Deuteronomy 21:15–17).

If there were no living sons and no descendants of any previously living sons, daughters inherit. In Numbers 27, the five daughters of Zelophehad come to Moses and ask for their father's inheritance, as they have no brothers. The order of inheritance is set out: a man's sons inherit first, daughters if no sons, brothers if he has no children, and so on.

Later, in Numbers 36, some of the heads of the families of the tribe of Manasseh come to Moses and point out that if a daughter inherits and then marries a man not from her paternal tribe, her land will pass from her birth tribe's inheritance into her marriage tribe's. So a further rule is laid down: if a daughter inherits land, she must marry someone within her father's tribe. (The daughters of Zelophehad marry the sons of their father's brothers. There is no indication that this was not their choice.)

The laws of Jewish inheritance are discussed in the Talmud, in the Mishneh Torah and by Saadiah ben Joseph among other sources.

Philo of Alexandria and Josephus also comment on the Jewish laws of inheritance, praising them above other law codes of their time. They also agreed that the firstborn son must receive a double portion of his father's estate.

Christian laws

At first, Christianity did not have its own inheritance traditions distinct from Judaism. With the accession of Emperor Constantine in 306, Christians began to distance themselves from Judaism and to exert influence over the laws and practices of secular institutions. From the beginning, this included inheritance. The Roman practice of adoption was a specific target because it was perceived as in conflict with the Judeo-Christian doctrine of primogeniture. As Stephanie Coontz documents in Marriage, a History (Penguin, 2006), not only succession but the whole constellation of rights and practices that included marriage, adoption, legitimacy, consanguinity, and inheritance changed in Western Europe from a Greco-Roman model to a Judeo-Christian pattern, based on Biblical and traditional Judeo-Christian principles. The transformation was essentially complete in the Middle Ages, although in English-speaking countries there was additional development under the influence of Protestantism. Even when Europe became secularized and Christianity faded into the background, the legal foundation Christendom had laid remained. Only in the era of modern jurisprudence have there been significant changes.

Islamic laws

Main article: Islamic inheritance jurisprudence

The Quran introduced some rights and restrictions regarding inheritance, including general improvements in the treatment of women and family life compared to the pre-Islamic societies of the Arabian Peninsula. Furthermore, the Quran introduced additional heirs that were not entitled to inheritance in pre-Islamic times, mentioning nine relatives specifically of which six were female and three were male. However, the inheritance rights of women remained different from those of men because in Islam, someone always has the responsibility of looking after a woman's expenses. According to 4:11, for example, a son is entitled to twice as much inheritance as a daughter. The Quran also presented efforts to fix the laws of inheritance, and thus forming a complete legal system. This development was in contrast to pre-Islamic societies, where rules of inheritance varied considerably. In addition to the above changes, the Quran imposed restrictions on testamentary powers of a Muslim in disposing their property.

Three verses of the Quran, 4:11, 4:12, and 4:176, give specific details on inheritance and shares, in addition to a few other verses dealing with testamentary matters. But this information was used as a starting point by Muslim jurists who expounded the laws of inheritance even further using Hadith, as well as methods of juristic reasoning like Qiyas. Nowadays, inheritance is considered an integral part of Sharia law and its application for Muslims is mandatory. However, many Muslim people (see Historical inheritance systems) follow other inheritance customs.

Inequality

Over time, wealth passes from generation to generation through inheritance. In 2024, the Silent Generation and baby boomers represented 25% of the population, but held 65% of all wealth in the US.

 

Older generations have accumulated more average wealth per person (vertical axis), but baby boomers as a group have the largest amount of wealth (areas within each rectangle). An estimated 73.9% of inheritors are already in the top ten percentiles of net worth.

Inheritance by amount and distribution received and action taken with inheritances in Great Britain between 2008 and 2010

The distribution of inherited wealth has varied greatly across cultures and legal traditions. In nations using civil law, for example, the right of children to inherit wealth from their parents in predefined ratios is enshrined in law. as far back as the Code of Hammurabi (ca. 1750 BC). In the US State of Louisiana, the only US state where the legal system is derived from the Napoleonic Code, this system is known as "forced heirship" which prohibits disinheritance of adult children except for a few narrowly defined reasons that a parent is obligated to prove. Other legal traditions, particularly in nations using common law, allow inheritances to be divided however one wishes, or to disinherit any child for any reason.

In cases of unequal inheritance, the majority might receive a small share while the minority receives a larger share. The amount of inheritance is often far less than the value of a business initially given to the son, especially when a son takes over a thriving multimillion-dollar business. Yet the daughter is given the balance of the actual inheritance, which amounts to far less than the value of the business initially given to the son. This is especially seen in old-world cultures, but continues in many families to this day.

Arguments for eliminating forced heirship include the right to property and the merit of individual allocation of capital over government wealth confiscation and redistribution, but this does not resolve what some^[who?] describe as the problem of unequal inheritance. In terms of inheritance inequality, some economists and sociologists focus on the intergenerational transmission of income or wealth, which is said to directly affect one's mobility (or immobility) and class position in society. Nations differ on the political structure and policy options that govern the transfer of wealth.

According to the American federal government statistics compiled by Mark Zandi in 1985, the average US inheritance was $39,000. In subsequent years, the total annual inheritance more than doubled, reaching nearly $200 billion. By 2050, an estimated $25 trillion in inheritance will be transmitted across generations.

Some researchers have attributed this rise to the baby boomers generation. Historically, the baby boomers were the largest influx of children conceived after World War II. For this reason, Thomas Shapiro suggests that this generation "is in the midst of benefiting from the greatest inheritance of wealth in history". Inherited wealth may help explain why many Americans who have become rich may have had a "substantial head start". In September 2012, according to the Institute for Policy Studies, "over 60 percent" of the Forbes richest 400 Americans "grew up in substantial privilege", and often (but not always) received substantial inheritances.

Other research has shown that many inheritances, large or small, are rapidly squandered. Similarly, analysis shows that over two-thirds of high-wealth families lose their wealth within two generations; almost 80% of high-wealth parents "feel the next generation is not financially responsible [and/or competent] enough to handle inheritance".

Social stratification

It has been argued that inheritance significantly affects social stratification. Inheritance is an integral component of family, economic, and legal institutions, and a basic mechanism of class stratification. It also affects the distribution of wealth at the societal level. The total cumulative effect of inheritance on stratification outcomes takes three forms, according to scholars who have examined the subject.

The first form of inheritance is the inheritance of cultural capital (i.e., linguistic styles, higher status social circles, and aesthetic preferences). The second form of inheritance is through familial interventions in the form of inter vivos transfers (i.e., gifts between the living), especially at crucial junctures in the life courses. Examples include milestones such as going to college, getting married, getting a job, and purchasing a home. The third form of inheritance is the transfer of bulk estates at the time of death of the testators, thus resulting in significant economic advantage accruing to their children. The average age of receiving an inheritance has been estimated at around 60 years. The origin of the stability of inequalities is material (personal possessions one can obtain) and is also cultural, rooted either in varying child-rearing practices that are geared to socialization according to social class and economic position. Child-rearing practices among those who inherit wealth may center around favoring some groups at the expense of others at the bottom of the social hierarchy.

Sociological and economic effects of inheritance inequality

It is further argued that the degree to which economic status and inheritance are transmitted across generations determines one's life chances in society. Although many have linked one's social origins and educational attainment to life chances and opportunities, education is not the most influential predictor of economic mobility. In fact, children of well-off parents generally receive better schooling and benefit from material, cultural, and genetic inheritances. Likewise, schooling attainment is often persistent across generations, and families with higher amounts of inheritance can acquire and transmit higher amounts of human capital. Lower amounts of human capital and inheritance can perpetuate inequality in the housing market and higher education. Research reveals that inheritance plays an important role in the accumulation of housing wealth. Those who receive an inheritance are more likely to own a home than those who do not, regardless of the size of the inheritance.

Often, racial or religious minorities and individuals from socially disadvantaged backgrounds receive less inheritance and wealth. As a result, mixed races might be excluded in inheritance privilege and are more likely to rent homes or live in poorer neighborhoods, as well as achieve lower educational attainment compared with whites in America.

Nations with the highest income and wealth inequalities often have the highest rates of homicide and disease (such as obesity, diabetes, and hypertension), which results in high mortality rates. A New York Times article reveals that the U.S. is the world's wealthiest nation, but "ranks twenty-ninth in life expectancy, right behind Jordan and Bosnia" and "has the second highest mortality rate of the comparable OECD countries". This has been attributed to the significant gap of inheritance inequality in the country, although there are clearly other factors, such as the affordability of healthcare.

When social and economic inequalities centered on inheritance are perpetuated by major social institutions such as the family, education, and religion, these differing life opportunities are argued to be transmitted across generations. As a result, this inequality is believed to become part of the overall social structure.

Women's unequal inheritance rights refer to the disparities and discriminatory practices that women face in inheriting property and assets compared to men. These inequalities stem from a combination of legal, cultural, and religious practices that often prioritize male heirs over female ones, resulting in significant socio-economic consequences for women.

Dynastic wealth

Dynastic wealth is monetary inheritance passed down to generations that did not earn it. Dynastic wealth is linked to the term Plutocracy. Much has been written about the rise and influence of dynastic wealth, including the bestselling book Capital in the Twenty-First Century by the French economist Thomas Piketty.

Bill Gates uses the term in his article "Why Inequality Matters".

Soviet response to inheritance

As Communism is founded on the Marxist Labor Theory of Value, any money collected in the course of a lifetime is justified if it was based on the fruits of the person's own labor and not from exploiting others. The first communist government installed after the Russian Revolution resolved to abolish the right of inheritance regardless of being the result of someone's work or exploitation, with some exceptions.

Taxation

Main article: Inheritance tax

Many states have inheritance taxes or estate taxes, under which a portion of any inheritance or estate becomes government revenue.

Inheritance and pensions

United Arab Emirates

In the United Arab Emirates, government pensions can, under specific conditions, be transferred to heirs upon a pensioner's death. This reflects a broader approach in some countries to support families of deceased retirees.