Kinetic energy

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Kinetic energy
The cars of a roller coaster reach their maximum kinetic energy when at the bottom of the path. When they start rising, the kinetic energy begins to be converted to gravitational potential energy. The sum of kinetic and potential energy in the system remains constant, ignoring losses due to rolling resistance and drag.

Émilie du Châtelet (1706–1749) was the first to publish the relation for kinetic energy ${\displaystyle E_{\text{kin}}\propto m{v}^2}$, derived from the experimental observation of objects dropped into clay. (Portrait by Maurice Quentin de La Tour.)

In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion.

In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is $\frac{1}{2}m{v}^2$.

The kinetic energy of an object is equal to the work, or force (F) in the direction of motion times its displacement (s), needed to accelerate the object from rest to its given speed. The same amount of work is done by the object when decelerating from its current speed to a state of rest.

The SI unit of energy is the joule, while the English unit of energy is the foot-pound.

In relativistic mechanics, $\frac{1}{2}m{v}^2$ is a good approximation of kinetic energy only when v is much less than the speed of light.

History and etymology

The adjective kinetic has its roots in the Greek word κίνησις kinesis, meaning "motion". The dichotomy between kinetic energy and potential energy can be traced back to Aristotle's concepts of actuality and potentiality.

The principle of classical mechanics that E ∝ mv² is conserved was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the living force or vis viva. Willem 's Gravesande of the Netherlands provided experimental evidence of this relationship in 1722. By dropping weights from different heights into a block of clay, Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet recognized the implications of the experiment and published an explanation.

The terms kinetic energy and work in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to Thomas Young, who in his 1802 lecture to the Royal Society, was the first to use the term energy to refer to kinetic energy in its modern sense, instead of vis viva. Gaspard-Gustave Coriolis published in 1829 the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson, later Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–1851. William Rankine, who had introduced the term "potential energy" in 1853, and the phrase "actual energy" to complement it, later cites William Thomson and Peter Tait as substituting the word "kinetic" for "actual".

Overview

Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. These can be categorized in two main classes: potential energy and kinetic energy. Kinetic energy is the movement energy of an object. Kinetic energy can be transferred between objects and transformed into other kinds of energy.

Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a cyclist transfers chemical energy provided by food to the bicycle and cyclist's store of kinetic energy as they increase their speed. On a level surface, this speed can be maintained without further work, except to overcome air resistance and friction. The chemical energy has been converted into kinetic energy, the energy of motion, but the process is not completely efficient and produces thermal energy within the cyclist.

The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling. The energy is not destroyed; it has only been converted to another form by friction. Alternatively, the cyclist could connect a dynamo to one of the wheels and generate some electrical energy on the descent. The bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as heat.

Like any physical quantity that is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Thus, the kinetic energy of an object is not invariant.

Spacecraft use chemical energy to launch and gain considerable kinetic energy to reach orbital velocity. In an entirely circular orbit, this kinetic energy remains constant because there is almost no friction in near-earth space. However, it becomes apparent at re-entry when some of the kinetic energy is converted to heat. If the orbit is elliptical or hyperbolic, then throughout the orbit kinetic and potential energy are exchanged; kinetic energy is greatest and potential energy lowest at closest approach to the earth or other massive body, while potential energy is greatest and kinetic energy the lowest at maximum distance. Disregarding loss or gain however, the sum of the kinetic and potential energy remains constant.

Kinetic energy can be passed from one object to another. In the game of billiards, the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down dramatically, and the ball it hit accelerates as the kinetic energy is passed on to it. Collisions in billiards are effectively elastic collisions, in which kinetic energy is preserved. In inelastic collisions, kinetic energy is dissipated in various forms of energy, such as heat, sound and binding energy (breaking bound structures).

Flywheels have been developed as a method of energy storage. This illustrates that kinetic energy is also stored in rotational motion.

Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula ⁠1/2⁠mv² given by classical mechanics is suitable. However, if the speed of the object is comparable to the speed of light, relativistic effects become significant and the relativistic formula is used. If the object is on the atomic or sub-atomic scale, quantum mechanical effects are significant, and a quantum mechanical model must be employed.

Kinetic energy for non-relativistic velocity

Treatments of kinetic energy depend upon the relative velocity of objects compared to the fixed speed of light. Speeds experienced directly by humans are non-relativisitic; higher speeds require the theory of relativity.

Kinetic energy of rigid bodies

In classical mechanics, the kinetic energy of a point object (an object so small that its mass can be assumed to exist at one point), or a non-rotating rigid body depends on the mass of the body as well as its speed. The kinetic energy is equal to half the product of the mass and the square of the speed. In formula form:

$${\displaystyle E_{\text{k}}=\frac{1}{2}m{v}^2}$$

where ${\displaystyle m}$ is the mass and ${\displaystyle v}$ is the speed (magnitude of the velocity) of the body. In SI units, mass is measured in kilograms, speed in metres per second, and the resulting kinetic energy is in joules.

For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as

$${\displaystyle E_{\text{k}}=\frac{1}{2}\cdot 80\text{kg}\cdot {\left(18\text{m/s}\right)}^2=12,960\text{J}=12.96\text{kJ}}$$

When a person throws a ball, the person does work on it to give it speed as it leaves the hand. The moving ball can then hit something and push it, doing work on what it hits. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e.,

$${\displaystyle Fs=\frac{1}{2}m{v}^2}$$

Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. For example, a car traveling twice as fast as another requires four times as much distance to stop, assuming a constant braking force. As a consequence of this quadrupling, it takes four times the work to double the speed.

The kinetic energy of an object is related to its momentum by the equation: $${\displaystyle E_{\text{k}}=\frac{p^2}{2m}}$$

where:

• ${\displaystyle p}$ is momentum
• ${\displaystyle m}$ is mass of the body

For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass ${\displaystyle m}$, whose center of mass is moving in a straight line with speed ${\displaystyle v}$, as seen above is equal to

$${\displaystyle E_{\text{t}}=\frac{1}{2}m{v}^2}$$

where:

• ${\displaystyle m}$ is the mass of the body
• ${\displaystyle v}$ is the speed of the center of mass of the body.

The kinetic energy of any entity depends on the reference frame in which it is measured. However, the total energy of an isolated system, i.e. one in which energy can neither enter nor leave, does not change over time in the reference frame in which it is measured. Thus, the chemical energy converted to kinetic energy by a rocket engine is divided differently between the rocket ship and its exhaust stream depending upon the chosen reference frame. This is called the Oberth effect. But the total energy of the system, including kinetic energy, fuel chemical energy, heat, etc., is conserved over time, regardless of the choice of reference frame. Different observers moving with different reference frames would however disagree on the value of this conserved energy.

The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the center of momentum frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the invariant mass of the system as a whole.

Derivation

Without vector calculus

The work W done by a force F on an object over a distance s parallel to F equals

$${\displaystyle W=F\cdot s.}$$

Using Newton's second law

$${\displaystyle F=ma}$$

with m the mass and a the acceleration of the object and

$${\displaystyle s=\frac{a{t}^2}{2}}$$

the distance traveled by the accelerated object in time t, we find with ${\displaystyle v=at}$ for the velocity v of the object

$${\displaystyle W=ma\frac{a{t}^2}{2}=\frac{m(at{)}^2}{2}=\frac{m{v}^2}{2}.}$$

With vector calculus

The work done in accelerating a particle with mass m during the infinitesimal time interval dt is given by the dot product of force F and the infinitesimal displacement dx

$${\displaystyle \mathbf{F}\cdot d\mathbf{x}=\mathbf{F}\cdot \mathbf{v}dt=\frac{d\mathbf{p}}{dt}\cdot \mathbf{v}dt=\mathbf{v}\cdot d\mathbf{p}=\mathbf{v}\cdot d(m\mathbf{v}),}$$

where we have assumed the relationship p = m v and the validity of Newton's second law. (However, also see the special relativistic derivation below.)

Applying the product rule we see that:

$${\displaystyle d(\mathbf{v}\cdot \mathbf{v})=(d\mathbf{v})\cdot \mathbf{v}+\mathbf{v}\cdot (d\mathbf{v})=2(\mathbf{v}\cdot d\mathbf{v}).}$$

Therefore (assuming constant mass so that dm = 0), we have

$${\displaystyle \mathbf{v}\cdot d(m\mathbf{v})=\frac{m}{2}d(\mathbf{v}\cdot \mathbf{v})=\frac{m}{2}d{v}^2=d\left(\frac{m{v}^2}{2}\right).}$$

Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy:

$${\displaystyle E_{\text{k}}={\int }_{v_1}^{v_2}\mathbf{p}\cdot d\mathbf{v}={\int }_{v_1}^{v_2}m\mathbf{v}\cdot d\mathbf{v}=\frac{m{v}^2}{2}{|}_{v_1}^{v_2}=\frac{1}{2}m(v_2^2-v_1^2).}$$

This equation states that the kinetic energy (E_k) is equal to the integral of the dot product of the momentum (p) of a body and the infinitesimal change of the velocity (v) of the body. It is assumed that the body starts with no kinetic energy when it is at rest (motionless).

Rotating bodies

If a rigid body Q is rotating about any line through the center of mass then it has rotational kinetic energy (${\displaystyle E_{\text{r}}}$) which is simply the sum of the kinetic energies of its moving parts, and is thus given by:

$${\displaystyle E_{\text{r}}={\int }_Q\frac{v^{2}dm}{2}={\int }_Q\frac{(r\omega {)}^{2}dm}{2}=\frac{{\omega }^2}{2}{\int }_Q{r}^{2}dm=\frac{{\omega }^2}{2}I=\frac{1}{2}I{\omega }^2}$$

where:

• ω is the body's angular velocity
• r is the distance of any mass dm from that line
• ${\displaystyle I}$ is the body's moment of inertia, equal to ${\int }_Q{r}^{2}dm$.

(In this equation the moment of inertia must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape).

Kinetic energy of systems

A system of bodies may have internal kinetic energy due to the relative motion of the bodies in the system. For example, in the Solar System the planets and planetoids are orbiting the Sun. In a tank of gas, the molecules are moving in all directions. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains.

A macroscopic body that is stationary (i.e. a reference frame has been chosen to correspond to the body's center of momentum) may have various kinds of internal energy at the molecular or atomic level, which may be regarded as kinetic energy, due to molecular translation, rotation, and vibration, electron translation and spin, and nuclear spin. These all contribute to the body's mass, as provided by the special theory of relativity. When discussing movements of a macroscopic body, the kinetic energy referred to is usually that of the macroscopic movement only. However, all internal energies of all types contribute to a body's mass, inertia, and total energy.

Fluid dynamics

In fluid dynamics, the kinetic energy per unit volume at each point in an incompressible fluid flow field is called the dynamic pressure at that point.

$${\displaystyle E_{\text{k}}=\frac{1}{2}m{v}^2}$$

Dividing by V, the unit of volume:

$${\displaystyle \begin{array}{rl}\frac{E_{\text{k}}}{V}& =\frac{1}{2}\frac{m}{V}{v}^2\\ q& =\frac{1}{2}\rho v^2\end{array}}$$

where ${\displaystyle q}$ is the dynamic pressure, and ρ is the density of the incompressible fluid.

Frame of reference

The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable inertial frame of reference. For example, a bullet passing an observer has kinetic energy in the reference frame of this observer. The same bullet is stationary to an observer moving with the same velocity as the bullet, and so has zero kinetic energy. By contrast, the total kinetic energy of a system of objects cannot be reduced to zero by a suitable choice of the inertial reference frame, unless all the objects have the same velocity. In any other case, the total kinetic energy has a non-zero minimum, as no inertial reference frame can be chosen in which all the objects are stationary. This minimum kinetic energy contributes to the system's invariant mass, which is independent of the reference frame.

The total kinetic energy of a system depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass.

This may be simply shown: let ${\displaystyle \mathbf{V}}$ be the relative velocity of the center of mass frame i in the frame k. Since

$${\displaystyle v^2={\left(v_i+V\right)}^2=\left({\mathbf{v}}_i+\mathbf{V}\right)\cdot \left({\mathbf{v}}_i+\mathbf{V}\right)={\mathbf{v}}_i\cdot {\mathbf{v}}_i+2{\mathbf{v}}_i\cdot \mathbf{V}+\mathbf{V}\cdot \mathbf{V}=v_i^2+2{\mathbf{v}}_i\cdot \mathbf{V}+V^2,}$$

Then,

$${\displaystyle E_{\text{k}}=\int \frac{v^2}{2}dm=\int \frac{v_i^2}{2}dm+\mathbf{V}\cdot \int {\mathbf{v}}_{i}dm+\frac{V^2}{2}\int dm.}$$

However, let $\int \frac{v_i^2}{2}dm=E_i$ the kinetic energy in the center of mass frame, $\int {\mathbf{v}}_{i}dm$ would be simply the total momentum that is by definition zero in the center of mass frame, and let the total mass: $\int dm=M$. Substituting, we get:

$${\displaystyle E_{\text{k}}=E_i+\frac{M{V}^2}{2}.}$$

Thus the kinetic energy of a system is lowest to center of momentum reference frames, i.e., frames of reference in which the center of mass is stationary (either the center of mass frame or any other center of momentum frame). In any different frame of reference, there is additional kinetic energy corresponding to the total mass moving at the speed of the center of mass. The kinetic energy of the system in the center of momentum frame is a quantity that is invariant (all observers see it to be the same).

Rotation in systems

It sometimes is convenient to split the total kinetic energy of a body into the sum of the body's center-of-mass translational kinetic energy and the energy of rotation around the center of mass (rotational energy):

$${\displaystyle E_{\text{k}}=E_{\text{t}}+E_{\text{r}}}$$

where:

• E_k is the total kinetic energy
• E_t is the translational kinetic energy
• E_r is the rotational energy or angular kinetic energy in the rest frame

Thus the kinetic energy of a tennis ball in flight is the kinetic energy due to its rotation, plus the kinetic energy due to its translation.

Relativistic kinetic energy

See also: Mass in special relativity and Tests of relativistic energy and momentum

If a body's speed is a significant fraction of the speed of light, it is necessary to use relativistic mechanics to calculate its kinetic energy. In relativity, the total energy is given by the energy-momentum relation:

$${\displaystyle E^2=(p\text{c}{)}^2+{\left(m_0{\text{c}}^2\right)}^2}$$

Here we use the relativistic expression for linear momentum: ${\displaystyle p=m\gamma v}$, where $\gamma =1/\sqrt{1-v^2/c^2}$. with ${\displaystyle m}$ being an object's (rest) mass, ${\displaystyle v}$ speed, and c the speed of light in vacuum. Then kinetic energy is the total relativistic energy minus the rest energy: $${\displaystyle E_K=E-m_0{c}^2=\sqrt{(p\text{c}{)}^2+{\left(m_0{\text{c}}^2\right)}^2}-m_0{c}^2}$$

At low speeds, the square root can be expanded and the rest energy drops out, giving the Newtonian kinetic energy.

Log of relativistic kinetic energy versus log relativistic momentum, for many objects of vastly different scales. The intersections of the object lines with the bottom axis approaches the rest energy. At low kinetic energy the slope of the object lines reflect Newtonian mechanics. As the lines approach ${\displaystyle c}$ the slope bends at the lightspeed barrier.

Derivation

Start with the expression for linear momentum ${\displaystyle \mathbf{p}=m\gamma \mathbf{v}}$, where $\gamma =1/\sqrt{1-v^2/c^2}$. Integrating by parts yields:$${\displaystyle E_{\text{k}}=\int \mathbf{v}\cdot d\mathbf{p}=\int \mathbf{v}\cdot d(m\gamma \mathbf{v})=m\gamma \mathbf{v}\cdot \mathbf{v}-\int m\gamma \mathbf{v}\cdot d\mathbf{v}=m\gamma v^2-\frac{m}{2}\int \gamma d\left(v^2\right)}$$Since ${\displaystyle \gamma ={\left(1-v^2/c^2\right)}^{-\frac{1}{2}}}$,$${\displaystyle \begin{array}{rl}{E}_{\text{k}}& =m\gamma v^2-\frac{-m{c}^2}{2}\int \gamma d\left(1-\frac{v^2}{c^2}\right)\\ & =m\gamma v^2+m{c}^2{\left(1-\frac{v^2}{c^2}\right)}^{\frac{1}{2}}-E_0\end{array}}$$Where ${\displaystyle E_0}$ is a constant of integration for the indefinite integral. Rearranging to combine the factor ${\displaystyle m\gamma }$ gives $${\displaystyle \begin{array}{rl}{E}_{\text{k}}& =m\gamma \left(v^2+c^2\left(1-\frac{v^2}{c^2}\right)\right)-E_0\\ & =m\gamma \left(v^2+c^2-v^2\right)-E_0\\ & =m\gamma c^2-E_0\end{array}}$$${\displaystyle E_0}$ is found by observing that when ${\displaystyle \mathbf{v}=0,\gamma =1}$ and ${\displaystyle E_{\text{k}}=0}$, the result is the "rest energy":

$${\displaystyle E_0=m{c}^2}$$

and resulting in the formula:

$${\displaystyle E_{\text{k}}=m\gamma c^2-m{c}^2=\frac{m{c}^2}{\sqrt{1-\frac{v^2}{c^2}}}-m{c}^2=(\gamma -1)m{c}^2}$$ This formula shows that the work expended accelerating an object from rest approaches infinity as the velocity approaches the speed of light. Thus it is impossible to accelerate an object across this boundary.

Low speed limit

The mathematical by-product of this calculation is the mass–energy equivalence formula, that mass and energy are essentially the same thing:

$${\displaystyle E_{\text{rest}}=E_0=m{c}^2}$$

At a low speed (v ≪ c), the relativistic kinetic energy is approximated well by the classical kinetic energy. To see this, apply the binomial approximation or take the first two terms of the Taylor expansion in powers of ${\displaystyle v^2}$ for the reciprocal square root:

$${\displaystyle E_{\text{k}}\approx m{c}^2\left(1+\frac{1}{2}\frac{v^2}{c^2}\right)-m{c}^2=\frac{1}{2}m{v}^2}$$

So, the total energy ${\displaystyle E_k}$ can be partitioned into the rest mass energy plus the non-relativistic kinetic energy at low speeds.

When objects move at a speed much slower than light (e.g. in everyday phenomena on Earth), the first two terms of the series predominate. The next term in the Taylor series approximation

$${\displaystyle E_{\text{k}}\approx m{c}^2\left(1+\frac{1}{2}\frac{v^2}{c^2}+\frac{3}{8}\frac{v^4}{c^4}\right)-m{c}^2=\frac{1}{2}m{v}^2+\frac{3}{8}m\frac{v^4}{c^2}}$$

is small for low speeds. For example, for a speed of 10 km/s (22,000 mph) the correction to the non-relativistic kinetic energy is 0.0417 J/kg (on a non-relativistic kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a non-relativistic kinetic energy of 5 GJ/kg).

The relativistic relation between kinetic energy and momentum is given by

$${\displaystyle E_{\text{k}}=\sqrt{p^2{c}^2+m^2{c}^4}-m{c}^2}$$

This can also be expanded as a Taylor series, the first term of which is the simple expression from Newtonian mechanics:

${\displaystyle E_{\text{k}}\approx \frac{p^2}{2m}-\frac{p^4}{8m^3{c}^2}.}$

This suggests that the formulae for energy and momentum are not special and axiomatic, but concepts emerging from the equivalence of mass and energy and the principles of relativity.

General relativity

See also: Schwarzschild geodesics

Using the convention that

$${\displaystyle g_{\alpha \beta }{u}^{\alpha }{u}^{\beta }=-c^2}$$

where the four-velocity of a particle is

$${\displaystyle u^{\alpha }=\frac{d{x}^{\alpha }}{d\tau }}$$

and ${\displaystyle \tau }$ is the proper time of the particle, there is also an expression for the kinetic energy of the particle in general relativity.

If the particle has momentum

$${\displaystyle p_{\beta }=m{g}_{\beta \alpha }{u}^{\alpha }}$$

as it passes by an observer with four-velocity u_obs, then the expression for total energy of the particle as observed (measured in a local inertial frame) is

$${\displaystyle E=-p_{\beta }{u}_{\text{obs}}^{\beta }}$$

and the kinetic energy can be expressed as the total energy minus the rest energy:

$${\displaystyle E_k=-p_{\beta }{u}_{\text{obs}}^{\beta }-m{c}^2.}$$

Consider the case of a metric that is diagonal and spatially isotropic (g_tt, g_ss, g_ss, g_ss). Since

$${\displaystyle u^{\alpha }=\frac{d{x}^{\alpha }}{dt}\frac{dt}{d\tau }=v^{\alpha }{u}^t}$$

where v^α is the ordinary velocity measured w.r.t. the coordinate system, we get

$${\displaystyle -c^2=g_{\alpha \beta }{u}^{\alpha }{u}^{\beta }=g_{tt}{\left(u^t\right)}^2+g_{ss}{v}^2{\left(u^t\right)}^2.}$$

Solving for u^t gives

$${\displaystyle u^t=c\sqrt{\frac{-1}{g_{tt}+g_{ss}{v}^2}}.}$$

Thus for a stationary observer (v = 0)

$${\displaystyle u_{\text{obs}}^t=c\sqrt{\frac{-1}{g_{tt}}}}$$

and thus the kinetic energy takes the form

$${\displaystyle E_{\text{k}}=-m{g}_{tt}{u}^t{u}_{\text{obs}}^t-m{c}^2=m{c}^2\sqrt{\frac{g_{tt}}{g_{tt}+g_{ss}{v}^2}}-m{c}^2.}$$

Factoring out the rest energy gives:

$${\displaystyle E_{\text{k}}=m{c}^2\left(\sqrt{\frac{g_{tt}}{g_{tt}+g_{ss}{v}^2}}-1\right).}$$

This expression reduces to the special relativistic case for the flat-space metric where

$${\displaystyle \begin{array}{rl}{g}_{tt}& =-c^2\\ g_{ss}& =1.\end{array}}$$

In the Newtonian approximation to general relativity

$${\displaystyle \begin{array}{rl}{g}_{tt}& =-\left(c^2+2\mathrm{\Phi }\right)\\ g_{ss}& =1-\frac{2\mathrm{\Phi }}{c^2}\end{array}}$$

where Φ is the Newtonian gravitational potential. This means clocks run slower and measuring rods are shorter near massive bodies.

Kinetic energy in quantum mechanics

Further information: Hamiltonian (quantum mechanics)

In quantum mechanics, observables like kinetic energy are represented as operators. For one particle of mass m, the kinetic energy operator appears as a term in the Hamiltonian and is defined in terms of the more fundamental momentum operator ${\displaystyle \hat{p}}$. The kinetic energy operator in the non-relativistic case can be written as

$${\displaystyle \hat{T}=\frac{{\hat{p}}^2}{2m}.}$$

Notice that this can be obtained by replacing ${\displaystyle p}$ by ${\displaystyle \hat{p}}$ in the classical expression for kinetic energy in terms of momentum,

$${\displaystyle E_{\text{k}}=\frac{p^2}{2m}.}$$

In the Schrödinger picture, ${\displaystyle \hat{p}}$ takes the form ${\displaystyle -i\hslash \mathrm{\nabla }}$ where the derivative is taken with respect to position coordinates and hence

$${\displaystyle \hat{T}=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2.}$$

The expectation value of the electron kinetic energy, ${\displaystyle ⟨\hat{T}⟩}$, for a system of N electrons described by the wavefunction ${\displaystyle |\psi ⟩}$ is a sum of 1-electron operator expectation values:

$${\displaystyle ⟨\hat{T}⟩=⟨\psi \left|\sum _{i=1}^N\frac{-{\hslash }^2}{2m_{\text{e}}}{\mathrm{\nabla }}_i^2\right|\psi ⟩=-\frac{{\hslash }^2}{2m_{\text{e}}}\sum _{i=1}^N⟨\psi \left|{\mathrm{\nabla }}_i^2\right|\psi ⟩}$$

where ${\displaystyle m_{\text{e}}}$ is the mass of the electron and ${\displaystyle {\mathrm{\nabla }}_i^2}$ is the Laplacian operator acting upon the coordinates of the i^th electron and the summation runs over all electrons.

The density functional formalism of quantum mechanics requires knowledge of the electron density only, i.e., it formally does not require knowledge of the wavefunction. Given an electron density ${\displaystyle \rho (\mathbf{r})}$, the exact N-electron kinetic energy functional is unknown; however, for the specific case of a 1-electron system, the kinetic energy can be written as

$${\displaystyle T[\rho ]=\frac{1}{8}\int \frac{\mathrm{\nabla }\rho (\mathbf{r})\cdot \mathrm{\nabla }\rho (\mathbf{r})}{\rho (\mathbf{r})}{d}^{3}r}$$

where ${\displaystyle T[\rho ]}$ is known as the von Weizsäcker kinetic energy functional.

Freud's psychoanalytic theories

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Freud%27s_psychoanalytic_theories

Sigmund Freud (c. 1921)

Sigmund Freud (6 May 1856 – 23 September 1939) is considered to be the founder of the psychodynamic approach to psychology, which looks to unconscious drives to explain human behavior. Freud believed that the mind is responsible for both conscious and unconscious decisions that it makes on the basis of psychological drives. The id, ego, and super-ego are three aspects of the mind Freud believed to comprise a person's personality. Freud believed people are "simply actors in the drama of [their] own minds, pushed by desire, pulled by coincidence. Underneath the surface, our personalities represent the power struggle going on deep within us".

Views on religion

Further information: Sigmund Freud's views on religion

Freud did not believe in the existence of a supernatural force that has pre-programmed us to behave in a certain way. His idea of the Id explains why people act out in certain ways when it is not in line with the ego or superego. "Religion is an illusion and it derives its strength from the fact that it falls in with our instinctual desires." Freud believed that people rely on religion to give explanations for anxieties and tension they do not want to consciously believe in. Freud argued that humanity created God in their image. This reverses the idea of any type of religion because he believed that it is constructed by the mind. The role of the mind is something that Freud repeatedly talked about because he believed that the mind is responsible for both conscious and unconscious decisions based on drives and forces. The idea that religion causes people to behave in a moral way is incorrect according to Freud because he believed that no other force has the power to control the ways in which people act. Unconscious desires motivate people to act accordingly.

Freud did a significant amount of research studying how people act and interact in a group setting. He believed that people act in different ways according to the demands and constraints of the group as a whole. In his book Group Psychology and the Analysis of the Ego, Freud argued that the church and organized religion form an "artificial group" which requires an external force to keep it together. In this type of group, everything is dependent on that external force, and without it, the group would no longer exist. Groups are necessary, according to Freud in order to decrease narcissism in all people, by creating libidinal ties with others by placing everyone at an equal level. The commonness among different people with different egos allows people to identify with one another. This relates to the idea of religion because Freud believed that people created religion in order to create these group ties that they unconsciously seek.

Oedipus complex

According to Freud's many theories of religion, the Oedipus complex is utilized in the understanding and mastery of religious beliefs. In Freud's psychosexual stages, he mentioned the Oedipus complex and how it affects children and their relationships with their same-sex parental figure. According to Freud, there is an unconscious desire for one's mother to be a virgin and for one's father to be an all-powerful, almighty figure. Freud's interest in Greek mythology and religion greatly influenced his psychological theories. The Oedipus complex is when a boy is jealous of his father. The boy strives to possess his mother and ultimately replace his father as a means of no longer having to fight for her undivided attention and affection. Along with seeking his mother's love, boys also experience castration anxiety which is the fear of losing their genitalia. Boys fear that their fathers will retaliate and castrate them as a result of desiring their mother. Females also experience penis envy which is the parallel reaction to the male experience of castration anxiety. Females are jealous of their fathers' penis and wish to have one as well. Girls then repress this feeling and instead long for a child of their own. This suppression leads to the girl identifying with her mother and acquiring feminine traits.

Psychoanalytic theory

Main article: Id, ego and super-ego

Psychoanalysis was founded by Sigmund Freud. Freud believed that people could be cured by making their motivations conscious. The aim of psychoanalysis therapy is to release repressed emotions and experiences, i.e., make the unconscious conscious. Psychoanalysis is commonly used to treat depression and anxiety disorders. It is only by having a cathartic (i.e., healing) experience that a person can be helped and "cured".

Id

The Id, according to Freud, is the part of the unconscious that seeks pleasure. His idea of the Id explains why people act out in certain ways when it is not in line with the ego or superego. The Id is the part of the mind which holds all of humankind's most basic and primal instincts. It is the impulsive, unconscious part of the mind that is based on the desire to seek immediate satisfaction. The Id does not have a grasp on any form of reality or consequence. Freud understood that some people are controlled by the Id because it makes people engage in need-satisfying behavior without any regard for what is right or wrong. Freud compared the Id and the Ego to a horse and a rider. The Id is compared to the horse, which is directed and controlled by the Ego, the rider. This example goes to show that although the Id is supposed to be controlled by the Ego, they often interact with one another according to the drives of the Ego. The Id is made up of two biological instincts, Eros which is the drive to create, and Thanatos which is the drive to destroy.

Ego

In order for people to maintain a realistic sense here on earth, the Ego is responsible for creating a balance between pleasure and pain. It is impossible for all desires of the Id to be met and the Ego realizes this but continues to seek pleasure and satisfaction. Although the Ego does not know the difference between right and wrong, it is aware that not all drives can be met at a given time. The reality principle is what the Ego operates on to help satisfy the Id’s demands while also compromising with the constraints of reality. The Ego is a person's "self" composed of unconscious desires. The Ego takes into account ethical and cultural ideals in order to balance out the desires originating in the Id. Although both the Id and the Ego are unconscious, the Ego has close contact with the perceptual system. The Ego has the function of self-preservation, which is why it has the ability to control the instinctual demands from the Id.

"The ego is first and foremost a bodily ego; it is not merely a surface entity but is itself the projection of a surface. If we wish to find an anatomical analogy for it we can best identify it with the ‘cortical homunculus’ of the anatomists, which stands on its head in the cortex, sticks up its heels, faces backwards and, as we know, has its speech-area on the left-hand side. The ego is ultimately derived from bodily sensations, chiefly from those springing from the surface of the body. It may thus be regarded as a mental projection of the surface of the body, representing the superficies of the mental apparatus."

Superego

The Superego, which develops around age four or five, incorporates the morals of society. Freud believed that the Superego is what allows the mind to control its impulses that are looked down upon morally. The Superego can be considered to be the conscience of the mind because it has the ability to distinguish between reality as well as what is right or wrong. Without the Superego, Freud believed people would act out with aggression and other immoral behaviors because the mind would have no way of understanding the difference between right and wrong. The Superego is considered to be the "consciousness" of a person's personality and can override the drives from the Id. Freud separates the Superego into two separate categories; the ideal self and the conscience. The conscience contains ideals and morals that exist within a society that prevent people from acting out based on their internal desires. The ideal self contains images of how people ought to behave according to society's ideals.

The unconscious

Main article: Unconscious mind

Freud believed that the answers to what controlled daily actions resided in the unconscious mind, despite alternative views that all our behaviors were conscious. He felt that religion is an illusion based on human values that are created by the mind to overcome inner psychological conflict. He believed that notions of the unconsciousness and gaps in the consciousness can be explained by acts of which the consciousness affords no evidence. The unconscious mind positions itself in every aspect of life, whether one is dormant or awake. Though one may be unaware of the impact of the unconscious mind, it influences the actions we engage in. Human behavior may be understood by searching for an analysis of mental processes. This explanation gives significance to verbal slips and dreams. They are caused by hidden reasons in the mind displayed in concealed forms. Verbal slips of the unconscious mind are referred to as a Freudian slip. This is a term to explain a spoken mistake derived from the unconscious mind. Traumatizing information on thoughts and beliefs is blocked from the conscious mind. Slips expose our true thoughts stored in the unconscious.

Sexual instincts or drives have deeply hidden roots in the unconscious mind. Instincts act by giving vitality and enthusiasm to the mind through meaning and purpose. The range of instincts is in great numbers. Freud expressed them in two categories. One is Eros, the self-preserving life instinct containing all erotic pleasures. While Eros is used for basic survival, the living instinct alone cannot explain all behavior, according to Freud. In contrast, Thanatos is the death instinct. It is full of self-destruction of sexual energy and our unconscious desire to die. The main part of human behavior and actions is tied back to sexual drives. Since birth, the existence of sexual drives can be recognized as one of the most important incentives of life.

Psychosexual stages

Main article: Psychosexual development

Freud's theory of psychosexual development is represented by five stages. According to Freud, each stage occurs within a specific time frame of one's life. If one becomes fixated in any of the five stages, he or she will develop personality traits that coincide with the specific stage and its focus.

• Oral Stage – The first stage is the oral stage. An infant is in this stage from birth to eighteen months of age. The main focus in the oral stage is pleasure-seeking through the infant's mouth. During this stage, the need for tasting and sucking becomes prominent in producing pleasure. Oral stimulation is crucial during this stage; if the infant's needs are not met during this time frame he or she will be fixated in the oral stage. Fixation in this stage can lead to adult habits such as thumb-sucking, smoking, over-eating, and nail-biting. Personality traits can also develop during adulthood that is linked to oral fixation; these traits can include optimism and independence or pessimism and hostility.
• Anal Stage – The second stage is the anal stage which lasts from eighteen months to three years of age. During this stage, the infant's pleasure-seeking centers are located in the bowels and bladder. Parents stress toilet training and bowel control during this time period. Fixation in the anal stage can lead to anal-retention or anal-expulsion. Anal retentive characteristics include being overly neat, precise, and orderly while being anal expulsive involves being disorganized, messy, and destructive.
• Phallic Stage – The third stage is the phallic stage. It begins at the age of three and continues until the age of six. Now, sensitivity becomes concentrated in the genitals and masturbation (in both sexes) becomes a new source of pleasure. The child becomes aware of anatomical sex differences, which sets in motion the conflict of jealousy and fear which Freud called the Oedipus complex (in boys). Later, the Freud scholars added the Electra complex (in girls).
• Latency Stage – The fourth stage is the latency stage which begins at the age of six and continues until the age of eleven. During this stage, there is no pleasure-seeking region of the body; instead, all sexual feelings are repressed. Thus, children are able to develop social skills and find comfort through peer and family interaction.
• Genital Stage – The final stage of psychosexual development is the genital stage. This stage starts from eleven onwards, lasts through puberty, and ends when one reaches adulthood at the age of eighteen. The onset of puberty reflects a strong interest from one person to another of the opposite sex. If one does not experience fixation in any of the psychosexual stages, once he or she has reached the genital stage, he or she will grow into a well-balanced human being.

Anxiety and defense mechanisms

Freud proposed a set of defense mechanisms in one's body. This set of defense mechanisms occurs so that one can hold a favorable or preferred view of oneself. For example, in a particular situation when an event occurs that violates one's preferred view of oneself, Freud stated that it is necessary for the self to have some mechanism to defend itself against this unfavorable event; this is known as a defense mechanism. Freud's work on defense mechanisms focused on how the ego defends itself against internal events or impulses, which are regarded as unacceptable to one's ego. These defense mechanisms are used to handle the conflict between the id, the ego, and the superego.

Freud noted that a major drive for people is the reduction of tension and the major cause of tension is anxiety. He identified three types of anxiety; reality anxiety, neurotic anxiety, and moral anxiety. Reality anxiety is the most basic form of anxiety and is based on the ego. It is typically based on the fear of real and possible events, for example, being bitten by a dog or falling off a roof. Neurotic anxiety comes from an unconscious fear that the basic impulses of the id will take control of the person, leading to eventual punishment for expressing the id's desires. Moral anxiety comes from the superego. It appears in the form of a fear of violating values or moral codes and appears as feelings like guilt or shame.

When anxiety occurs, the mind's first response is to seek rational ways of escaping the situation by increasing problem-solving efforts and a range of defense mechanisms may be triggered. These are ways that the ego develops to help deal with the id and the superego. Defense mechanisms often appear unconsciously and tend to distort or falsify reality. When the distortion of reality occurs, there is a change in perception which allows for a lessening in anxiety, resulting in a reduction of the tension one experiences. Sigmund Freud noted a number of ego defenses that were noted throughout his work, but his daughter, Anna Freud, developed and elaborated on them. The defense mechanisms are as follows: 1) Denial is believing that what is true is actually false 2) Displacement is taking out impulses on a less threatening target 3) Intellectualization is avoiding unacceptable emotions by focusing on the intellectual aspects 4) Projection is attributing uncomfortable feelings to others 5) Rationalization is creating false but believable justifications 6) Reaction Formation is taking the opposite belief because the true belief causes anxiety 7) Regression is going back to a previous stage of development 8) Repression is pushing uncomfortable thoughts out of conscious awareness 9) Suppression is consciously forcing unwanted thoughts out of our awareness 10) Sublimation is redirecting ‘wrong’ urges into socially acceptable actions. These defenses are not under our conscious control and our unconscious will use one or more to protect ourselves from stressful situations. They are natural and normal and without these, neurosis develops, such as anxiety states, phobias, obsessions, or hysteria.

Totem and Taboo

Totem and Taboo

Freud desired to understand religion and spirituality and dealt with the nature of religious beliefs in many of his books and essays. He regarded God as an illusion, based on the infantile need for a powerful father figure. Freud believed that religion was an expression of underlying psychological neuroses and distress. In some of his writing, he suggested that religion is an attempt to control the Oedipal complex, as he goes on to discuss in his book Totem and Taboo.

In 1913, Freud published the book, Totem and Taboo. This book was an attempt to reconstruct the birth and the process of development of religion as a social institution. He wanted to demonstrate how the study of psychoanalysis is important in the understanding of the growth of civilization. This book is about how the Oedipus complex, which is when an infant develops an attachment for the mother early on in life, and the incest taboo came into being and why they are present in all human societies. The incest taboo rises because of a desire for incest. The purpose of the totemic animal is not for group unity, but to reinforce the incest taboo. The totemic animal is not a symbol of God but a symbol of the father and it is an important part of religious development. Totemism originates from the memory of an event in pre-history where the male group members eat the father figure due to a desire for the females. The guilt they feel for their actions and for the loss of a father figure leads them to prohibit incest in a new way. Totemism is a means of preventing incest and as a ritual reminder of the murder of the father. This shows that sexual desire, since there are many social prohibitions on sexual relations, is channeled through certain ritual actions and all societies adapt these rituals so that sexuality develops in approved ways. This reveals unconscious desires and their repression. Freud believes that civilization makes people unhappy because it contradicts the desire for progress, freedom, happiness, and wealth. Civilization requires the repression of drives and instincts such as sexual, aggression, and the death instinct in order that civilization can work.

According to Freud, religion originated in prehistoric collective experiences that became repressed and ritualized as totems and taboos. He stated that most, if not all religions, can be traced back to early human sacrifice including Christianity in which Christ on the cross is a symbolic representation of killing the father and eating the father figure is shown with ‘the body of Christ’, also known as Communion. In this work, Freud attributed the origin of religion to emotions such as hatred, fear, and jealousy. These emotions are directed towards the father figure in the clan from the sons who are denied sexual desires towards the females. Freud attributed totem religions to be a result of extreme emotion, rash action, and the result of guilt.

The Psychopathology of Everyday Life

The Psychopathology of Everyday Life

The Psychopathology of Everyday Life is one of the most important books in psychology. It was written by Freud in 1901, and it laid the basis for the theory of psychoanalysis. The book contains twelve chapters on forgetting things such as names, childhood memories, mistakes, clumsiness, slips of the tongue, and determinism of the unconscious. Freud believed that there were reasons that people forget things like words, names, and memories. He also believed that mistakes in speech, now referred to as a Freudian Slip, were not accidents but instead the "dynamic unconscious" revealing something meaningful.

Freud suggested that our everyday psychopathology is a minor disturbance of mental life which may quickly pass away. Freud believed all of these acts to have an important significance; the most trivial slips of the tongue or pen may reveal people's secret feelings and fantasies. Pathology is brought into everyday life which Freud pointed out through dreams, forgetfulness, and parapraxes. He used these things to make his case for the existence of an unconscious that refuses to be explained or contained by consciousness. Freud explained how the forgetting of multiple events in our everyday life can be a consequence of repression, suppression, denial, displacement, and identification. Defense mechanisms occur to protect one's ego so in The Psychopathology of Everyday Life, Freud stated, "painful memories merge into motivated forgetting which special ease". (p. 154)

Three Essays on the Theory of Sexuality

Three Essays on the Theory of Sexuality, sometimes titled Three Contributions to the Theory of Sex, written in 1905 by Sigmund Freud explores and analyzes his theory of sexuality and its presence throughout childhood. Freud's book describes three main topics in reference to sexuality: sexual perversions, childhood sexuality, and puberty. His first essay in this series is called "The Sexual Aberrations." This essay focuses on the distinction between a sexual object and a sexual aim. A sexual object is an object that one desires, while the sexual aim is the acts that one desires to perform with the object. Freud's second essay is titled "Infantile Sexuality." In this essay, he insists that children have sexual urges. The psychosexual stages are the steps a child must take in order to continue having sexual urges once adulthood is reached. The third essay Freud wrote describes "The Transformation of Puberty." In this essay, he examines how children express their sexuality throughout puberty and how sexual identity is formed during this time frame. Freud ultimately attempted to link unconscious sexual desires to conscious actions in each of his essays.

The Interpretation of Dreams

The Interpretation of Dreams is one of Sigmund Freud's best-known published works. It set the stage for his psychoanalytic work and Freud's approach to the unconscious with regard to the interpretation of dreams. During therapy sessions with patients, Freud would ask his patients to discuss what was on their minds. Frequently, the responses were directly related to a dream. As a result, Freud began to analyze dreams, believing that it gave him access to one's deepest thoughts. In addition, he was able to find links between one's current hysterical behaviors and past traumatic experiences. From these experiences, he began to write a book that was designed to help others to understand dream interpretation. In the book, he discussed his theory of the unconscious.

Freud believed that dreams were messages from the unconscious masked as wishes controlled by internal stimuli. The unconscious mind plays the most imperative role in dream interpretation. In order to remain in a state of sleep, the unconscious mind has to suppress negative thoughts and represent them in any edited form. Therefore, when one dreams, the unconscious makes an effort to deal with conflict. It would enable one to begin to act on them.

There are four steps required to convert dreams from latent or unconscious thoughts to the manifest content. They are condensation, displacement, symbolism, and secondary revision. Ideas first go through a process of condensation that takes thoughts and turns them into a single image. Then, the true emotional meaning of the dream loses its significance in an element of displacement. This is followed by symbolism representing our latent thoughts in visual form. A special focus on symbolism was emphasized in the interpretation of dreams. Our dreams are highly symbolic with an underlying principle meaning. Many of the symbolic stages focus on sexual connotations. For example, a tree branch could represent a penis. Freud believed all human behavior originated from our sexual drives and desires. In the last stage of converting dreams to manifest content, dreams are made sensible. The final product of manifest content is what we remember when we come out of our sleep.