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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 (?S = 0) under small changes in the configuration of the system (?q).
where is the Lagrangianfunction for the system. In other words, any first-order perturbation of the true evolution results in (at most) second-order changes in . The action is a functional, i.e., something that takes as its input a function and returns a single number, a scalar. In terms of functional analysis, Hamilton's principle states that the true evolution of a physical system is a solution of the functional equation
That is, the system takes a path in configuration space for which the action is stationary, with fixed boundary conditions at the beginning and the end of the path.
Euler-Lagrange equations derived from the action integral
Requiring that the true trajectory q(t) be a stationary point of the action functional is equivalent to a set of differential equations for q(t) (the Euler-Lagrange equations), which may be derived as follows.
Let q(t) represent the true evolution of the system between two specified states q1 = q(t1) and q2 = q(t2) at two specified times t1 and t2, and let ?(t) be a small perturbation that is zero at the endpoints of the trajectory
To first order in the perturbation ?(t), the change in the action functional would be
where we have expanded the LagrangianL to first order in the perturbation ?(t).
The boundary conditions causes the first term to vanish
Hamilton's principle requires that this first-order change is zero for all possible perturbations ?(t), i.e., the true path is a stationary point of the action functional (either a minimum, maximum or saddle point). This requirement can be satisfied if and only if
These equations are called the Euler-Lagrange equations for the variational problem.
Canonical momenta and constants of motion
The conjugate momentumpk for a generalized coordinate qk is defined by the equation
An important special case of the Euler-Lagrange equation occurs when L does not contain a generalized coordinate qk explicitly,
that is, the conjugate momentum is a constant of the motion.
In such cases, the coordinate qk is called a cyclic coordinate. For example, if we use polar coordinates t, r, ? to describe the planar motion of a particle, and if L does not depend on ?, the conjugate momentum is the conserved angular momentum.
Example: Free particle in polar coordinates
Trivial examples help to appreciate the use of the action principle via the Euler-Lagrange equations. A free particle (mass m and velocity v) in Euclidean space moves in a straight line. Using the Euler-Lagrange equations, this can be shown in polar coordinates as follows. In the absence of a potential, the Lagrangian is simply equal to the kinetic energy
in orthonormal (x,y) coordinates, where the dot represents differentiation with respect to the curve parameter (usually the time, t). Therefore, upon application of the Euler-Lagrange equations,
And likewise for y. Thus the Euler-Lagrange formulation can be used to derive Newton's laws.
In polar coordinates (r, ?) the kinetic energy and hence the Lagrangian becomes
The radial r and ? components of the Euler-Lagrange equations become, respectively
The solution of these two equations is given by
for a set of constants a, b, c, d determined by initial conditions.
Thus, indeed, the solution is a straight line given in polar coordinates: a is the velocity, c is the distance of the closest approach to the origin, and d is the angle of motion.
Applied to deformable bodies
Hamilton's principle is an important variational principle in elastodynamics. As opposed to a system composed of rigid bodies, deformable bodies have an infinite number of degrees of freedom and occupy continuous regions of space; consequently, the state of the system is described by using continuous functions of space and time. The extended Hamilton Principle for such bodies is given by
where T is the kinetic energy, U is the elastic energy, We is the work done by
external loads on the body, and t1, t2 the initial and final times. If the system is conservative, the work done by external forces may be derived from a scalar potential V. In this case,
This is called Hamilton's principle and it is invariant under coordinate transformations.
where p = (p1, p2, ..., pN) are the conjugate momenta defined above. By contrast, Hamilton's principle uses , the integral of the Lagrangian over time.
the solution that they determine...
Hamilton's principle determines the trajectory q(t) as a function of time, whereas Maupertuis' principle determines only the shape of the trajectory in the generalized coordinates. For example, Maupertuis' principle determines the shape of the ellipse on which a particle moves under the influence of an inverse-square central force such as gravity, but does not describe per se how the particle moves along that trajectory. (However, this time parameterization may be determined from the trajectory itself in subsequent calculations using the conservation of energy). By contrast, Hamilton's principle directly specifies the motion along the ellipse as a function of time.
...and the constraints on the variation.
Maupertuis' principle requires that the two endpoint states q1 and q2 be given and that energy be conserved along every trajectory (same energy for each trajectory). This forces the endpoint times to be varied as well. By contrast, Hamilton's principle does not require the conservation of energy, but does require that the endpoint times t1 and t2 be specified as well as the endpoint states q1 and q2.
The path of a body in a gravitational field (i.e. free fall in space time, a so-called geodesic) can be found using the action principle.
Quantum mechanics and quantum field theory
In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all imaginable paths and the value of their action. The action corresponding to the various paths is used to calculate the path integral, that gives the probability amplitudes of the various outcomes.
Although equivalent in classical mechanics with Newton's laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. In particular, it is fully appreciated and best understood within quantum mechanics. Richard Feynman's path integral formulation of quantum mechanics is based on a stationary-action principle, using path integrals. Maxwell's equations can be derived as conditions of stationary action.