Analytical mechanics

From Wikipedia, the free encyclopedia

In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics.

By contrast, analytical mechanics uses scalar properties of motion representing the system as a whole—usually its total kinetic energy and potential energy—not Newton's vectorial forces of individual particles. A scalar is a quantity, whereas a vector is represented by quantity and direction. The equations of motion are derived from the scalar quantity by some underlying principle about the scalar's variation.

Analytical mechanics takes advantage of a system's constraints to solve problems. The constraints limit the degrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as generalized coordinates. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics allows numerous mechanical problems to be solved with greater efficiency than fully vectorial methods. It does not always work for non-conservative forces or dissipative forces like friction, in which case one may revert to Newtonian mechanics or use the Udwadia–Kalaba equation.

Two dominant branches of analytical mechanics are Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities in configuration space) and Hamiltonian mechanics (using coordinates and corresponding momenta in phase space). Both formulations are equivalent by a Legendre transformation on the generalized coordinates, velocities and momenta, therefore both contain the same information for describing the dynamics of a system. There are other formulations such as Hamilton–Jacobi theory, Routhian mechanics, and Appell's equation of motion. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called the principle of least action. One result is Noether's theorem, a statement which connects conservation laws to their associated symmetries.

Analytical mechanics does not introduce new physics and is not more general than Newtonian mechanics. Rather it is a collection of equivalent formalisms which have broad application. In fact the same principles and formalisms can be used in relativistic mechanics and general relativity, and with some modification, quantum mechanics and quantum field theory.

Analytical mechanics is used widely, from fundamental physics to applied mathematics, particularly chaos theory.

The methods of analytical mechanics apply to discrete particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom. The definitions and equations have a close analogy with those of mechanics.

Intrinsic motion

Generalized coordinates and constraints

In Newtonian mechanics, one customarily uses all three Cartesian coordinates, or other 3D coordinate system, to refer to a body's position during its motion. In physical systems, however, some structure or other system usually constrains the body's motion from taking certain directions and pathways. So a full set of Cartesian coordinates is often unneeded, as the constraints determine the evolving relations among the coordinates, which relations can be modeled by equations corresponding to the constraints. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motion's geometry, reducing the number of coordinates to the minimum needed to model the motion.
These are known as generalized coordinates, denoted q_i (i = 1, 2, 3...).

Difference between curvillinear and generalized coordinates

Generalized coordinates incorporate constraints on the system. There is one generalized coordinate q_i for each degree of freedom (for convenience labelled by an index i = 1, 2...N), i.e. each way the system can change its configuration; as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates. The number of curvilinear coordinates equals the dimension of the position space in question (usually 3 for 3d space), while the number of generalized coordinates is not necessarily equal to this dimension; constraints can reduce the number of degrees of freedom (hence the number of generalized coordinates required to define the configuration of the system), following the general rule:

[dimension of position space (usually 3)] × [number of constituents of system ("particles")] − (number of constraints)

= (number of degrees of freedom) = (number of generalized coordinates)

For a system with N degrees of freedom, the generalized coordinates can be collected into an N-tuple:

${\displaystyle \mathbf {q} =(q_{1},q_{2},\cdots q_{N})}$

and the time derivative (here denoted by an overdot) of this tuple give the generalized velocities:

${\displaystyle {\frac {d\mathbf {q} }{dt}}=\left({\frac {dq_{1}}{dt}},{\frac {dq_{2}}{dt}},\cdots {\frac {dq_{N}}{dt}}\right)\equiv \mathbf {\dot {q}} =({\dot {q}}_{1},{\dot {q}}_{2},\cdots {\dot {q}}_{N})}$.

D'Alembert's principle

The foundation which the subject is built on is D'Alembert's principle.

This principle states that infinitesimal virtual work done by a force across reversible displacements is zero, which is the work done by a force consistent with ideal constraints of the system. The idea of a constraint is useful - since this limits what the system can do, and can provide steps to solving for the motion of the system. The equation for D'Alembert's principle is:

${\displaystyle \delta W={\boldsymbol {\mathcal {Q}}}\cdot \delta \mathbf {q} =0\,,}$

where

${\displaystyle {\boldsymbol {\mathcal {Q}}}=({\mathcal {Q}}_{1},{\mathcal {Q}}_{2},\cdots {\mathcal {Q}}_{N})}$

are the generalized forces (script Q instead of ordinary Q is used here to prevent conflict with canonical transformations below) and q are the generalized coordinates. This leads to the generalized form of Newton's laws in the language of analytical mechanics:

${\displaystyle {\boldsymbol {\mathcal {Q}}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial T}{\partial \mathbf {\dot {q}} }}\right)-{\frac {\partial T}{\partial \mathbf {q} }}\,,}$

where T is the total kinetic energy of the system, and the notation

${\displaystyle {\frac {\partial }{\partial \mathbf {q} }}=\left({\frac {\partial }{\partial q_{1}}},{\frac {\partial }{\partial q_{2}}},\cdots {\frac {\partial }{\partial q_{N}}}\right)}$

is a useful shorthand (see matrix calculus for this notation).

Holonomic constraints

If the curvilinear coordinate system is defined by the standard position vector r, and if the position vector can be written in terms of the generalized coordinates q and time t in the form:

${\displaystyle \mathbf {r} =\mathbf {r} (\mathbf {q} (t),t)}$

and this relation holds for all times t, then q are called Holonomic constraints. Vector r is explicitly dependent on t in cases when the constraints vary with time, not just because of q(t). For time-independent situations, the constraints are also called scleronomic, for time-dependent cases they are called rheonomic.

Lagrangian mechanics

Lagrangian and Euler–Lagrange equations

The introduction of generalized coordinates and the fundamental Lagrangian function:

${\displaystyle L(\mathbf {q} ,\mathbf {\dot {q}} ,t)=T(\mathbf {q} ,\mathbf {\dot {q}} ,t)-V(\mathbf {q} ,\mathbf {\dot {q}} ,t)}$

where T is the total kinetic energy and V is the total potential energy of the entire system, then either following the calculus of variations or using the above formula - lead to the Euler–Lagrange equations;

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right)={\frac {\partial L}{\partial \mathbf {q} }}\,,}$

which are a set of N second-order ordinary differential equations, one for each q_i(t).

This formulation identifies the actual path followed by the motion as a selection of the path over which the time integral of kinetic energy is least, assuming the total energy to be fixed, and imposing no conditions on the time of transit.

Configuration space

The Lagrangian formulation uses the configuration space of the system, the set of all possible generalized coordinates:

${\displaystyle {\mathcal {C}}=\{\mathbf {q} \in \mathbb {R} ^{N}\}\,,}$

where ${\displaystyle \mathbb {R} ^{N}}$ is N-dimensional real space. The particular solution to the Euler–Lagrange equations is called a (configuration) path or trajectory, i.e. one particular q(t) subject to the required initial conditions. The general solutions form a set of possible configurations as functions of time:

${\displaystyle \{\mathbf {q} (t)\in \mathbb {R} ^{N}\,:\,t\geq 0,t\in \mathbb {R} \}\subseteq {\mathcal {C}}\,,}$

The configuration space can be defined more generally, and indeed more deeply, in terms of topological manifolds and the tangent bundle.

Hamiltonian mechanics

Hamiltonian and Hamilton's equations

The Legendre transformation of the Lagrangian replaces the generalized coordinates and velocities (q, q̇) with (q, p); the generalized coordinates and the generalized momenta conjugate to the generalized coordinates:

${\displaystyle \mathbf {p} ={\frac {\partial L}{\partial \mathbf {\dot {q}} }}=\left({\frac {\partial L}{\partial {\dot {q}}_{1}}},{\frac {\partial L}{\partial {\dot {q}}_{2}}},\cdots {\frac {\partial L}{\partial {\dot {q}}_{N}}}\right)=(p_{1},p_{2}\cdots p_{N})\,,}$

and introduces the Hamiltonian (which is in terms of generalized coordinates and momenta):

${\displaystyle H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {\dot {q}} ,t)}$

where • denotes the dot product, also leading to Hamilton's equations:

${\displaystyle \mathbf {\dot {p}} =-{\frac {\partial H}{\partial \mathbf {q} }}\,,\quad \mathbf {\dot {q}} =+{\frac {\partial H}{\partial \mathbf {p} }}\,,}$

which are now a set of 2N first-order ordinary differential equations, one for each q_i(t) and p_i(t). Another result from the Legendre transformation relates the time derivatives of the Lagrangian and Hamiltonian:

${\displaystyle {\frac {dH}{dt}}=-{\frac {\partial L}{\partial t}}\,,}$

which is often considered one of Hamilton's equations of motion additionally to the others. The generalized momenta can be written in terms of the generalized forces in the same way as Newton's second law:

${\displaystyle \mathbf {\dot {p}} ={\boldsymbol {\mathcal {Q}}}\,.}$

Generalized momentum space

Analogous to the configuration space, the set of all momenta is the momentum space (technically in this context; generalized momentum space):

${\displaystyle {\mathcal {M}}=\{\mathbf {p} \in \mathbb {R} ^{N}\}\,.}$

"Momentum space" also refers to "k-space"; the set of all wave vectors (given by De Broglie relations) as used in quantum mechanics and theory of waves: this is not referred to in this context.

Phase space

The set of all positions and momenta form the phase space;

${\displaystyle {\mathcal {P}}={\mathcal {C}}\times {\mathcal {M}}=\{(\mathbf {q} ,\mathbf {p} )\in \mathbb {R} ^{2N}\}\,,}$

that is, the Cartesian product × of the configuration space and generalized momentum space.

A particular solution to Hamilton's equations is called a phase path, a particular curve (q(t),p(t)) subject to the required initial conditions. The set of all phase paths, the general solution to the differential equations, is the phase portrait:

${\displaystyle \{(\mathbf {q} (t),\mathbf {p} (t))\in \mathbb {R} ^{2N}\,:\,t\geq 0,t\in \mathbb {R} \}\subseteq {\mathcal {P}}\,,}$

The Poisson bracket

All dynamical variables can be derived from position r, momentum p, and time t, and written as a function of these: A = A(q, p, t). If A(q, p, t) and B(q, p, t) are two scalar valued dynamical variables, the Poisson bracket is defined by the generalized coordinates and momenta:

${\displaystyle {\begin{aligned}\{A,B\}\equiv \{A,B\}_{\mathbf {q} ,\mathbf {p} }&={\frac {\partial A}{\partial \mathbf {q} }}\cdot {\frac {\partial B}{\partial \mathbf {p} }}-{\frac {\partial A}{\partial \mathbf {p} }}\cdot {\frac {\partial B}{\partial \mathbf {q} }}\\&\equiv \sum _{k}{\frac {\partial A}{\partial q_{k}}}{\frac {\partial B}{\partial p_{k}}}-{\frac {\partial A}{\partial p_{k}}}{\frac {\partial B}{\partial q_{k}}}\,,\end{aligned}}}$

Calculating the total derivative of one of these, say A, and substituting Hamilton's equations into the result leads to the time evolution of A:

${\displaystyle {\frac {dA}{dt}}=\{A,H\}+{\frac {\partial A}{\partial t}}\,.}$

This equation in A is closely related to the equation of motion in the Heisenberg picture of quantum mechanics, in which classical dynamical variables become quantum operators (indicated by hats (^)), and the Poisson bracket is replaced by the commutator of operators via Dirac's canonical quantization:

${\displaystyle \{A,B\}\rightarrow {\frac {1}{i\hbar }}[{\hat {A}},{\hat {B}}]\,.}$

Properties of the Lagrangian and Hamiltonian functions

Following are overlapping properties between the Lagrangian and Hamiltonian functions.

• All the individual generalized coordinates q_i(t), velocities q̇_i(t) and momenta p_i(t) for every degree of freedom are mutually independent. Explicit time-dependence of a function means the function actually includes time t as a variable in addition to the q(t), p(t), not simply as a parameter through q(t) and p(t), which would mean explicit time-independence.
• The Lagrangian is invariant under addition of the total time derivative of any function of q and t, that is:

${\displaystyle L'=L+{\frac {d}{dt}}F(\mathbf {q} ,t)\,,}$

so each Lagrangian L and L' describe exactly the same motion. In other words, the Lagrangian of a system is not unique.

• Analogously, the Hamiltonian is invariant under addition of the partial time derivative of any function of q, p and t, that is:

${\displaystyle K=H+{\frac {\partial }{\partial t}}G(\mathbf {q} ,\mathbf {p} ,t)\,,}$

(K is a frequently used letter in this case). This property is used in canonical transformations (see below).

• If the Lagrangian is independent of some generalized coordinates, then the generalized momenta conjugate to those coordinates are constants of the motion, i.e. are conserved, this immediately follows from Lagrange's equations:

${\displaystyle {\frac {\partial L}{\partial q_{j}}}=0\,\rightarrow \,{\frac {dp_{j}}{dt}}={\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}=0}$

Such coordinates are "cyclic" or "ignorable". It can be shown that the Hamiltonian is also cyclic in exactly the same generalized coordinates.

• If the Lagrangian is time-independent the Hamiltonian is also time-independent (i.e. both are constant in time).
• If the kinetic energy is a homogeneous function of degree 2 of the generalized velocities, and the Lagrangian is explicitly time-independent, then:

${\displaystyle T((\lambda {\dot {q}}_{i})^{2},(\lambda {\dot {q}}_{j}\lambda {\dot {q}}_{k}),\mathbf {q} )=\lambda ^{2}T(({\dot {q}}_{i})^{2},{\dot {q}}_{j}{\dot {q}}_{k},\mathbf {q} )\,,\quad L(\mathbf {q} ,\mathbf {\dot {q}} )\,,}$

where λ is a constant, then the Hamiltonian will be the total conserved energy, equal to the total kinetic and potential energies of the system:

${\displaystyle H=T+V=E\,.}$

This is the basis for the Schrödinger equation, inserting quantum operators directly obtains it.

Principle of least action

As the system evolves, q traces a path through configuration space (only some are shown). The path taken by the system (red) has a stationary action (δS = 0) under small changes in the configuration of the system (δq).

Action is another quantity in analytical mechanics defined as a functional of the Lagrangian:

${\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)dt\,.}$

A general way to find the equations of motion from the action is the principle of least action:

${\displaystyle \delta {\mathcal {S}}=\delta \int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)dt=0\,,}$

where the departure t₁ and arrival t₂ times are fixed. The term "path" or "trajectory" refers to the time evolution of the system as a path through configuration space ${\displaystyle {\mathcal {C}}}$, in other words q(t) tracing out a path in ${\displaystyle {\mathcal {C}}}$. The path for which action is least is the path taken by the system.

From this principle, all equations of motion in classical mechanics can be derived. This approach can be extended to fields rather than a system of particles (see below), and underlies the path integral formulation of quantum mechanics, and is used for calculating geodesic motion in general relativity.

Hamiltonian-Jacobi mechanics

Canonical transformations

The invariance of the Hamiltonian (under addition of the partial time derivative of an arbitrary function of p, q, and t) allows the Hamiltonian in one set of coordinates q and momenta p to be transformed into a new set Q = Q(q, p, t) and P = P(q, p, t), in four possible ways:

${\displaystyle {\begin{aligned}&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{1}(\mathbf {q} ,\mathbf {Q} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{2}(\mathbf {q} ,\mathbf {P} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{3}(\mathbf {p} ,\mathbf {Q} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{4}(\mathbf {p} ,\mathbf {P} ,t)\\\end{aligned}}}$

With the restriction on P and Q such that the transformed Hamiltonian system is:

${\displaystyle \mathbf {\dot {P}} =-{\frac {\partial K}{\partial \mathbf {Q} }}\,,\quad \mathbf {\dot {Q}} =+{\frac {\partial K}{\partial \mathbf {P} }}\,,}$

the above transformations are called canonical transformations, each function G_n is called a generating function of the "nth kind" or "type-n". The transformation of coordinates and momenta can allow simplification for solving Hamilton's equations for a given problem.

The choice of Q and P is completely arbitrary, but not every choice leads to a canonical transformation. One simple criterion for a transformation q → Q and p → P to be canonical is the Poisson bracket be unity,

${\displaystyle \{Q_{i},P_{i}\}=1}$

for all i = 1, 2,...N. If this does not hold then the transformation is not canonical.

The Hamilton–Jacobi equation

By setting the canonically transformed Hamiltonian K = 0, and the type-2 generating function equal to Hamilton's principal function (also the action ${\displaystyle {\mathcal {S}}}$) plus an arbitrary constant C:

${\displaystyle G_{2}(\mathbf {q} ,t)={\mathcal {S}}(\mathbf {q} ,t)+C\,,}$

the generalized momenta become:

${\displaystyle \mathbf {p} ={\frac {\partial {\mathcal {S}}}{\partial \mathbf {q} }}}$

and P is constant, then the Hamiltonian-Jacobi equation (HJE) can be derived from the type-2 canonical transformation:

${\displaystyle H=-{\frac {\partial {\mathcal {S}}}{\partial t}}}$

where H is the Hamiltonian as before:

${\displaystyle H=H(\mathbf {q} ,\mathbf {p} ,t)=H\left(\mathbf {q} ,{\frac {\partial {\mathcal {S}}}{\partial \mathbf {q} }},t\right)}$

Another related function is Hamilton's characteristic function

${\displaystyle W(\mathbf {q} )={\mathcal {S}}(\mathbf {q} ,t)+Et}$

used to solve the HJE by additive separation of variables for a time-independent Hamiltonian H.

The study of the solutions of the Hamilton–Jacobi equations leads naturally to the study of symplectic manifolds and symplectic topology. In this formulation, the solutions of the Hamilton–Jacobi equations are the integral curves of Hamiltonian vector fields.

Routhian mechanics

Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, not often used but especially useful for removing cyclic coordinates. If the Lagrangian of a system has s cyclic coordinates q = q₁, q₂, ... q_s with conjugate momenta p = p₁, p₂, ... p_s, with the rest of the coordinates non-cyclic and denoted ζ = ζ₁, ζ₁, ..., ζ_{N − s}, they can be removed by introducing the Routhian:

${\displaystyle R=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {p} ,{\boldsymbol {\zeta }},{\dot {\boldsymbol {\zeta }}})\,,}$

which leads to a set of 2s Hamiltonian equations for the cyclic coordinates q,

${\displaystyle {\dot {\mathbf {q} }}=+{\frac {\partial R}{\partial \mathbf {p} }}\,,\quad {\dot {\mathbf {p} }}=-{\frac {\partial R}{\partial \mathbf {q} }}\,,}$

and N − s Lagrangian equations in the non cyclic coordinates ζ.

${\displaystyle {\frac {d}{dt}}{\frac {\partial R}{\partial {\dot {\boldsymbol {\zeta }}}}}={\frac {\partial R}{\partial {\boldsymbol {\zeta }}}}\,.}$

Set up in this way, although the Routhian has the form of the Hamiltonian, it can be thought of a Lagrangian with N − s degrees of freedom.

The coordinates q do not have to be cyclic, the partition between which coordinates enter the Hamiltonian equations and those which enter the Lagrangian equations is arbitrary. It is simply convenient to let the Hamiltonian equations remove the cyclic coordinates, leaving the non cyclic coordinates to the Lagrangian equations of motion.

Appellian mechanics

Appell's equation of motion involve generalized accelerations, the second time derivatives of the generalized coordinates:

${\displaystyle \alpha _{r}={\ddot {q}}_{r}={\frac {d^{2}q_{r}}{dt^{2}}}\,,}$

as well as generalized forces mentioned above in D'Alembert's principle. The equations are

${\displaystyle {\mathcal {Q}}_{r}={\frac {\partial S}{\partial \alpha _{r}}}\,,\quad S={\frac {1}{2}}\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}^{2}\,,}$

where

${\displaystyle \mathbf {a} _{k}={\ddot {\mathbf {r} }}_{k}={\frac {d^{2}\mathbf {r} _{k}}{dt^{2}}}}$

is the acceleration of the k particle, the second time derivative of its position vector. Each acceleration a_k is expressed in terms of the generalized accelerations α_r, likewise each r_k are expressed in terms the generalized coordinates q_r.

Extensions to classical field theory

Lagrangian field theory

Generalized coordinates apply to discrete particles. For N scalar fields φ_i(r, t) where i = 1, 2, ... N, the Lagrangian density is a function of these fields and their space and time derivatives, and possibly the space and time coordinates themselves:

${\displaystyle {\mathcal {L}}={\mathcal {L}}(\phi _{1},\phi _{2},\ldots \nabla \phi _{1},\nabla \phi _{2},\ldots \partial \phi _{1}/\partial t,\partial \phi _{2}/\partial t,\ldots \mathbf {r} ,t)\,.}$

and the Euler–Lagrange equations have an analogue for fields:

${\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{i})}}\right)={\frac {\partial {\mathcal {L}}}{\partial \phi _{i}}}\,,}$

where ∂_μ denotes the 4-gradient and the summation convention has been used. For N scalar fields, these Lagranian field equations are a set of N second order partial differential equations in the fields, which in general will be coupled and nonlinear.

This scalar field formulation can be extended to vector fields, tensor fields, and spinor fields.

The Lagrangian is the volume integral of the Lagrangian density:

${\displaystyle L=\int _{\mathcal {V}}{\mathcal {L}}\,dV\,.}$

Originally developed for classical fields, the above formulation is applicable to all physical fields in classical, quantum, and relativistic situations: such as Newtonian gravity, classical electromagnetism, general relativity, and quantum field theory. It is a question of determining the correct Lagrangian density to generate the correct field equation.

Hamiltonian field theory

The corresponding "momentum" field densities conjugate to the N scalar fields φ_i(r, t) are:

${\displaystyle \pi _{i}(\mathbf {r} ,t)={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}_{i}}}\,\quad {\dot {\phi }}_{i}\equiv {\frac {\partial \phi _{i}}{\partial t}}.}$

where in this context the overdot denotes a partial time derivative, not a total time derivative. The Hamiltonian density ${\displaystyle {\mathcal {H}}}$ is defined by analogy with mechanics:

${\displaystyle {\mathcal {H}}(\phi _{1},\phi _{2},\ldots ,\pi _{1},\pi _{2},\ldots ,\mathbf {r} ,t)=\sum _{i=1}^{N}{\dot {\phi }}_{i}(\mathbf {r} ,t)\pi _{i}(\mathbf {r} ,t)-{\mathcal {L}}\,.}$

The equations of motion are:

${\displaystyle {\dot {\phi }}_{i}=+{\frac {\delta {\mathcal {H}}}{\delta \pi _{i}}}\,,\quad {\dot {\pi }}_{i}=-{\frac {\delta {\mathcal {H}}}{\delta \phi _{i}}}\,,}$

where the variational derivative

${\displaystyle {\frac {\delta }{\delta \phi _{i}}}={\frac {\partial }{\partial \phi _{i}}}-\partial _{\mu }{\frac {\partial }{\partial (\partial _{\mu }\phi _{i})}}}$

must be used instead of merely partial derivatives. For N fields, these Hamiltonian field equations are a set of 2N first order partial differential equations, which in general will be coupled and nonlinear.

Again, the volume integral of the Hamiltonian density is the Hamiltonian

${\displaystyle H=\int _{\mathcal {V}}{\mathcal {H}}\,dV\,.}$

Symmetry, conservation, and Noether's theorem

Symmetry transformations in classical space and time

Each transformation can be described by an operator (i.e. function acting on the position r or momentum p variables to change them). The following are the cases when the operator does not change r or p, i.e. symmetries.

Transformation	Operator	Position	Momentum
Translational symmetry	${\displaystyle X(\mathbf {a} )}$	${\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\mathbf {a} }$	${\displaystyle \mathbf {p} \rightarrow \mathbf {p} }$
Time translation	${\displaystyle U(t_{0})}$	${\displaystyle \mathbf {r} (t)\rightarrow \mathbf {r} (t+t_{0})}$	${\displaystyle \mathbf {p} (t)\rightarrow \mathbf {p} (t+t_{0})}$
Rotational invariance	${\displaystyle R(\mathbf {\hat {n}} ,\theta )}$	${\displaystyle \mathbf {r} \rightarrow R(\mathbf {\hat {n}} ,\theta )\mathbf {r} }$	${\displaystyle \mathbf {p} \rightarrow R(\mathbf {\hat {n}} ,\theta )\mathbf {p} }$
Galilean transformations	${\displaystyle G(\mathbf {v} )}$	${\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\mathbf {v} t}$	${\displaystyle \mathbf {p} \rightarrow \mathbf {p} +m\mathbf {v} }$
Parity	${\displaystyle P}$	${\displaystyle \mathbf {r} \rightarrow -\mathbf {r} }$	${\displaystyle \mathbf {p} \rightarrow -\mathbf {p} }$
T-symmetry	${\displaystyle T}$	${\displaystyle \mathbf {r} \rightarrow \mathbf {r} (-t)}$	${\displaystyle \mathbf {p} \rightarrow -\mathbf {p} (-t)}$

where R(n̂, θ) is the rotation matrix about an axis defined by the unit vector n̂ and angle θ.

Noether's theorem

Noether's theorem states that a continuous symmetry transformation of the action corresponds to a conservation law, i.e. the action (and hence the Lagrangian) doesn't change under a transformation parameterized by a parameter s:

${\displaystyle L[q(s,t),{\dot {q}}(s,t)]=L[q(t),{\dot {q}}(t)]}$

the Lagrangian describes the same motion independent of s, which can be length, angle of rotation, or time. The corresponding momenta to q will be conserved.

Inertia

From Wikipedia, the free encyclopedia

Inertia is the resistance, of any physical object, to any change in its velocity. This includes changes to the object's speed, or direction of motion.

An aspect of this property is the tendency of objects to keep moving in a straight line at a constant speed, when no forces are upon them — and this aspect in particular is also called inertia.

The principle of inertia is one of the fundamental principles in classical physics that are still used today to describe the motion of objects and how they are affected by the applied forces on them.

Inertia comes from the Latin word, iners, meaning idle, sluggish. Inertia is one of the primary manifestations of mass, which is a quantitative property of physical systems. Isaac Newton defined inertia as his first law in his Philosophiæ Naturalis Principia Mathematica, which states: 

The vis insita, or innate force of matter, is a power of resisting by which every body, as much as in it lies, endeavours to preserve its present state, whether it be of rest or of moving uniformly forward in a straight line.

In common usage, the term "inertia" may refer to an object's "amount of resistance to change in velocity" (which is quantified by its mass), or sometimes to its momentum, depending on the context. The term "inertia" is more properly understood as shorthand for "the principle of inertia" as described by Newton in his First Law of Motion: an object not subject to any net external force moves at a constant velocity. Thus, an object will continue moving at its current velocity until some force causes its speed or direction to change.

On the surface of the Earth, inertia is often masked by the effects of friction and air resistance, both of which tend to decrease the speed of moving objects (commonly to the point of rest), and gravity. This misled the philosopher Aristotle to believe that objects would move only as long as force was applied to them:

...it [body] stops when the force which is pushing the travelling object has no longer power to push it along...

History and development of the concept

Early understanding of motion

Prior to the Renaissance, the most generally accepted theory of motion in Western philosophy was based on Aristotle who around about 335 BC to 322 BC said that, in the absence of an external motive power, all objects (on Earth) would come to rest and that moving objects only continue to move so long as there is a power inducing them to do so. Aristotle explained the continued motion of projectiles, which are separated from their projector, by the action of the surrounding medium, which continues to move the projectile in some way. Aristotle concluded that such violent motion in a void was impossible.

Despite its general acceptance, Aristotle's concept of motion was disputed on several occasions by notable philosophers over nearly two millennia. For example, Lucretius (following, presumably, Epicurus) stated that the "default state" of matter was motion, not stasis. In the 6th century, John Philoponus criticized the inconsistency between Aristotle's discussion of projectiles, where the medium keeps projectiles going, and his discussion of the void, where the medium would hinder a body's motion. Philoponus proposed that motion was not maintained by the action of a surrounding medium, but by some property imparted to the object when it was set in motion. Although this was not the modern concept of inertia, for there was still the need for a power to keep a body in motion, it proved a fundamental step in that direction. This view was strongly opposed by Averroes and by many scholastic philosophers who supported Aristotle. However, this view did not go unchallenged in the Islamic world, where Philoponus did have several supporters who further developed his ideas.

Theory of impetus

In the 14th century, Jean Buridan rejected the notion that a motion-generating property, which he named impetus, dissipated spontaneously. Buridan's position was that a moving object would be arrested by the resistance of the air and the weight of the body which would oppose its impetus. Buridan also maintained that impetus increased with speed; thus, his initial idea of impetus was similar in many ways to the modern concept of momentum. Despite the obvious similarities to more modern ideas of inertia, Buridan saw his theory as only a modification to Aristotle's basic philosophy, maintaining many other peripatetic views, including the belief that there was still a fundamental difference between an object in motion and an object at rest. Buridan also believed that impetus could be not only linear, but also circular in nature, causing objects (such as celestial bodies) to move in a circle.

Buridan's thought was followed up by his pupil Albert of Saxony (1316–1390) and the Oxford Calculators, who performed various experiments that further undermined the classical, Aristotelian view. Their work in turn was elaborated by Nicole Oresme who pioneered the practice of demonstrating laws of motion in the form of graphs.

Shortly before Galileo's theory of inertia, Giambattista Benedetti modified the growing theory of impetus to involve linear motion alone:

"…[Any] portion of corporeal matter which moves by itself when an impetus has been impressed on it by any external motive force has a natural tendency to move on a rectilinear, not a curved, path."

Benedetti cites the motion of a rock in a sling as an example of the inherent linear motion of objects, forced into circular motion.

Classical inertia

Galileo Galilei

The principle of inertia which originated with Aristotle for "motions in a void" states that an object tends to resist a change in motion. According to Newton, an object will stay at rest or stay in motion (i.e. "maintain its velocity") unless acted on by a net external force, whether it results from gravity, friction, contact, or some other force. The Aristotelian division of motion into mundane and celestial became increasingly problematic in the face of the conclusions of Nicolaus Copernicus in the 16th century, who argued that the earth (and everything on it) was in fact never "at rest", but was actually in constant motion around the sun. Galileo, in his further development of the Copernican model, recognized these problems with the then-accepted nature of motion and, at least partially as a result, included a restatement of Aristotle's description of motion in a void as a basic physical principle:

A body moving on a level surface will continue in the same direction at a constant speed unless disturbed.

Galileo writes that "all external impediments removed, a heavy body on a spherical surface concentric with the earth will maintain itself in that state in which it has been; if placed in movement towards the west (for example), it will maintain itself in that movement." This notion which is termed "circular inertia" or "horizontal circular inertia" by historians of science, is a precursor to, but distinct from, Newton's notion of rectilinear inertia. For Galileo, a motion is "horizontal" if it does not carry the moving body towards or away from the centre of the earth, and for him, "a ship, for instance, having once received some impetus through the tranquil sea, would move continually around our globe without ever stopping."

It is also worth noting that Galileo later (in 1632) concluded that based on this initial premise of inertia, it is impossible to tell the difference between a moving object and a stationary one without some outside reference to compare it against. This observation ultimately came to be the basis for Einstein to develop the theory of Special Relativity.

The first physicist to completely break away from the Aristotelian model of motion was Isaac Beeckman in 1614.

Concepts of inertia in Galileo's writings would later come to be refined, modified and codified by Isaac Newton as the first of his Laws of Motion (first published in Newton's work, Philosophiae Naturalis Principia Mathematica, in 1687):

Unless acted upon by a net unbalanced force, an object will maintain a constant velocity.

Note that "velocity" in this context is defined as a vector, thus Newton's "constant velocity" implies both constant speed and constant direction (and also includes the case of zero speed, or no motion). Since initial publication, Newton's Laws of Motion (and by inclusion, this first law) have come to form the basis for the branch of physics known as classical mechanics.

The term "inertia" was first introduced by Johannes Kepler in his Epitome Astronomiae Copernicanae (published in three parts from 1617–1621); however, the meaning of Kepler's term (which he derived from the Latin word for "idleness" or "laziness") was not quite the same as its modern interpretation. Kepler defined inertia only in terms of a resistance to movement, once again based on the presumption that rest was a natural state which did not need explanation. It was not until the later work of Galileo and Newton unified rest and motion in one principle that the term "inertia" could be applied to these concepts as it is today.

Nevertheless, despite defining the concept so elegantly in his laws of motion, even Newton did not actually use the term "inertia" to refer to his First Law. In fact, Newton originally viewed the phenomenon he described in his First Law of Motion as being caused by "innate forces" inherent in matter, which resisted any acceleration. Given this perspective, and borrowing from Kepler, Newton attributed the term "inertia" to mean "the innate force possessed by an object which resists changes in motion"; thus, Newton defined "inertia" to mean the cause of the phenomenon, rather than the phenomenon itself. However, Newton's original ideas of "innate resistive force" were ultimately problematic for a variety of reasons, and thus most physicists no longer think in these terms. As no alternate mechanism has been readily accepted, and it is now generally accepted that there may not be one which we can know, the term "inertia" has come to mean simply the phenomenon itself, rather than any inherent mechanism. Thus, ultimately, "inertia" in modern classical physics has come to be a name for the same phenomenon described by Newton's First Law of Motion, and the two concepts are now considered to be equivalent.

Relativity

Albert Einstein's theory of special relativity, as proposed in his 1905 paper entitled "On the Electrodynamics of Moving Bodies" was built on the understanding of inertia and inertial reference frames developed by Galileo and Newton. While this revolutionary theory did significantly change the meaning of many Newtonian concepts such as mass, energy, and distance, Einstein's concept of inertia remained unchanged from Newton's original meaning (in fact, the entire theory was based on Newton's definition of inertia). However, this resulted in a limitation inherent in special relativity: the principle of relativity could only apply to reference frames that were inertial in nature (meaning when no acceleration was present). In an attempt to address this limitation, Einstein proceeded to develop his general theory of relativity ("The Foundation of the General Theory of Relativity," 1916), which ultimately provided a unified theory for both inertial and noninertial (accelerated) reference frames. However, in order to accomplish this, in general relativity, Einstein found it necessary to redefine several fundamental concepts (such as gravity) in terms of a new concept of "curvature" of space-time, instead of the more traditional system of forces understood by Newton.

As a result of this redefinition, Einstein also redefined the concept of "inertia" in terms of geodesic deviation instead, with some subtle but significant additional implications. The result of this is that, according to general relativity, inertia is the gravitational coupling between matter and spacetime.

When dealing with very large scales, the traditional Newtonian idea of "inertia" does not actually apply and cannot necessarily be relied upon. Luckily, for sufficiently small regions of spacetime, the special theory can be used and inertia still means the same (and works the same) as in the classical model.

Another profound conclusion of the theory of special relativity—perhaps the most well known—was that energy and mass are not separate things but are, in fact, interchangeable. But this new relationship also carried with it new implications for the concept of inertia. The logical conclusion of special relativity was that if mass exhibits the principle of inertia, then inertia must also apply to energy. This theory, and subsequent experiments confirming some of its conclusions, have also served to radically expand the meaning of inertia to apply more widely and to include inertia of energy.

Interpretations

Mass and inertia

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Suspended weight showing inertia. The weight is suspended on a string, and has another string hanging from it. If the string is pulled quickly, only the bottom string is detached, and the weight remains hanging because of its inability to react in time (its inertness). If the same bottom string is pulled more slowly, the entire assembly falls down.

Physicists and mathematicians appear to be less inclined to use the popular concept of inertia as "a tendency to maintain momentum" and instead favor the mathematically useful definition of inertia as the measure of a body's resistance to changes in velocity or simply a body's inertial mass.

This was clear at the beginning of the 20th century, before the advent of the theory of relativity. Mass, m, denoted something like an amount of substance or quantity of matter. At the same time, mass was the quantitative measure of inertia of a body.

The mass of a body determines the momentum, ${\displaystyle p}$, of the body at given velocity, ${\displaystyle v}$; it is a proportionality factor in the formula:

${\displaystyle p=mv}$

The factor m is referred to as inertial mass.

But mass, as related to the "inertia" of a body, can also be defined by the formula:

${\displaystyle F=ma,}$

where F is the force, m is the inertial mass, and a is the acceleration.

By this formula, the greater its mass, the less a body accelerates under a given force. The masses defined by the above formulas are equal because the latter formula is a consequence of the former, if mass does not depend on time and velocity. Thus, "mass is the quantitative or numerical measure of a body’s inertia, that is of its resistance to being accelerated".

This meaning of a body's inertia therefore is altered from the popular meaning as "a tendency to maintain momentum" to a description of the measure of how difficult it is to change the velocity of a body, but it is consistent with the fact that motion in one reference frame can disappear in another, so it is the change in velocity that is important.

Inertial mass

There is no measurable difference between gravitational mass and inertial mass. The gravitational mass is defined by the quantity of gravitational field material a mass possesses, including its energy. The "inertial mass" (relativistic mass) is a function of the acceleration a mass has undergone and its resultant speed. A mass that has been accelerated to speeds close to the speed of light has its "relativistic mass" increased, and that is why the magnetic field strength in particle accelerators must be increased to force the mass's path to curve. In practice, "inertial mass" is normally taken to be "invariant mass" and so is identical to gravitational mass without the energy component.

Gravitational mass is measured by comparing the force of gravity of an unknown mass to the force of gravity of a known mass. This is typically done with some sort of balance. Equal masses will match on a balance because the gravitational field applies to them equally, producing identical weight. This assumption breaks down near supermassive objects such as black holes and neutron stars due to tidal effects. It also breaks down in weightless environments, because no matter what objects are compared, it will yield a balanced reading.

Inertial mass is found by applying a known net force to an unknown mass, measuring the resulting acceleration, and applying Newton's Second Law, ${\displaystyle F=ma}$. This gives an accurate value for mass, limited only by the accuracy of the measurements. When astronauts need to be measured in the weightlessness of free fall, they actually find their inertial mass in a special chair called a body mass measurement device (BMMD).

At high speeds, and especially near the speed of light (${\displaystyle c=2.99792458\cdot 10^{8}\,\mathrm {m/s} }$), inertial mass can be determined by measuring the magnetic field strength and the curvature of the path of an electrically-charged mass such as an electron.

No physical difference has been found between gravitational and inertial mass in a given inertial frame. In experimental measurements, the two always agree within the margin of error for the experiment. Einstein used the fact that gravitational and inertial mass were equal to begin his general theory of relativity, in which he postulated that gravitational mass was the same as inertial mass, and that the acceleration of gravity is a result of a "valley" or slope in the space-time continuum that masses "fell down". Dennis Sciama later showed that the reaction force produced by the combined gravity of all matter in the universe upon an accelerating object is mathematically equal to the object's inertia, but this would only be a workable physical explanation if, by some mechanism, the gravitational effects operated instantaneously.

At any non-zero speed, relativistic mass always exceeds gravitational mass. If the mass is made to travel close to the speed of light, its "inertial mass" (relativistic) as observed from a stationary frame would be very great while its gravitational mass would remain at its rest value, but the gravitational effect of the extra energy would exactly balance the measured increase in inertial mass.

Inertial frames

In a location such as a steadily moving railway carriage, a dropped ball (as seen by an observer in the carriage) would behave as it would if it were dropped in a stationary carriage. The ball would simply descend vertically. It is possible to ignore the motion of the carriage by defining it as an inertial frame. In a moving but non-accelerating frame, the ball behaves normally because the train and its contents continue to move at a constant velocity. Before being dropped, the ball was traveling with the train at the same speed, and the ball's inertia ensured that it continued to move in the same speed and direction as the train, even while dropping. Note that, here, it is inertia which ensured that, not its mass.

In an inertial frame, all the observers in uniform (non-accelerating) motion will observe the same laws of physics. However, observers in another inertial frame can make a simple, and intuitively obvious, transformation (the Galilean transformation), to convert their observations. Thus, an observer from outside the moving train could deduce that the dropped ball within the carriage fell vertically downwards.

However, in reference frames which are experiencing acceleration (non-inertial reference frames), objects appear to be affected by fictitious forces. For example, if the railway carriage were accelerating, the ball would not fall vertically within the carriage but would appear to an observer to be deflected because the carriage and the ball would not be traveling at the same speed while the ball was falling. Other examples of fictitious forces occur in rotating frames such as the earth. For example, a missile at the North Pole could be aimed directly at a location and fired southwards. An observer would see it apparently deflected away from its target by a force (the Coriolis force), but in reality, the southerly target has moved because earth has rotated while the missile is in flight. Because the earth is rotating, a useful inertial frame of reference is defined by the stars, which only move imperceptibly during most observations. Newton's first law of motion is known as the principle of inertia.

In summary, the principle of inertia is intimately linked with the principles of conservation of energy and conservation of momentum.

Rotational inertia

Another form of inertia is rotational inertia (→ moment of inertia), the property that a rotating rigid body maintains its state of uniform rotational motion. Its angular momentum is unchanged, unless an external torque is applied; this is also called conservation of angular momentum. Rotational inertia depends on the object remaining structurally intact as a rigid body, and also has practical consequences. For example, a gyroscope uses the property that it resists any change in the axis of rotation.

A gas or liquid in a container will also resist changes in rotational rate.