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The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th century) of finding an analogy between the propagation of light and the motion of a particle. The wave equation followed by mechanical systems is similar to, but not identical with, Schrödinger's equation, as described below; for this reason, the HJE is considered the "closest approach" of classical mechanics to quantum mechanics.
The conjugate momenta correspond to the first derivatives of S with respect to the generalized coordinates
As a solution to the Hamilton-Jacobi equation, the principal function contains N + 1 undetermined constants, the first N of them denoted as ?1, ?2 ... ?N, and the last one coming from the integration of .
are also constants of motion, and these equations can be inverted to find q as a function of all the ? and ? constants and time.
Comparison with other formulations of mechanics
The HJE is a single, first-order partial differential equation for the function S of the Ngeneralized coordinatesq1...qN and the time t. The generalized momenta do not appear, except as derivatives of S. Remarkably, the function S is equal to the classical action.
For comparison, in the equivalent Euler-Lagrange equations of motion of Lagrangian mechanics, the conjugate momenta also do not appear; however, those equations are a system of N, generally second-order equations for the time evolution of the generalized coordinates. Similarly, Hamilton's equations of motion are another system of 2N first-order equations for the time evolution of the generalized coordinates and their conjugate momenta p1...pN.
The HJE establishes a duality between trajectories and wave fronts. For example, in geometrical optics, light can be considered either as "rays" or waves. The wave front can be defined as the surface that the light emitted at time has reached at time . Light rays and wave fronts are dual: if one is known, the other can be deduced.
More precisely, geometrical optics is a variational problem where the "action" is the travel time along a path:
where is the index of the medium and is an infinitesimal arc length. From the above formulation, one can compute the ray paths using the Euler-Lagrange formulation; alternatively, one can compute the wave fronts by solving the Hamilton-Jacobi equation. Knowing one leads to knowing the other.
The above duality is very general and applies to all systems that derive from a variational principle: either compute the trajectories using Euler-Lagrange equations or the wave fronts by using Hamilton-Jacobi equation.
The wave front at time , for a system initially at at time , is defined as the collection of points such that . If is known, the momentum is immediately deduced:
Once is known, tangents to the trajectories are computed by solving the equation
for , where is the Lagrangian. The trajectories are then recovered from the knowledge of .
and Hamilton's equations in terms of the new variables P, Q and new Hamiltonian K have the same form:
To derive the HJE, we choose a generating function G2(q, P, t) in such a way that, it will make the new Hamiltonian K = 0.
Hence, all its derivatives are also zero, and the transformed Hamilton's equations become trivial
so the new generalized coordinates and momenta are constants of motion. As they are constants, in this context the new generalized momenta P are usually denoted ?1, ?2 ... ?N, i.e. Pm = ?m, and the new generalized coordinatesQ are typically denoted as ?1, ?2 ... ?N, so Qm = ?m.
Setting the generating function equal to Hamilton's principal function, plus an arbitrary constant A:
the HJE automatically arises:
Once we have solved for S(q, ?, t), these also give us the useful equations
or written in components for clarity
Ideally, these N equations can be inverted to find the original generalized coordinatesq as a function of the constants ?, ? and t, thus solving the original problem.
Action and Hamilton's functions
Hamilton's principal function S and classical function H are both closely related to action. The total differential of S is:
The HJE is most useful when it can be solved via additive separation of variables, which directly identifies constants of motion. For example, the time t can be separated if the Hamiltonian does not depend on time explicitly. In that case, the time derivative in the HJE must be a constant, usually denoted (-E), giving the separated solution
where the time-independent function W(q) is sometimes called Hamilton's characteristic function. The reduced Hamilton-Jacobi equation can then be written
To illustrate separability for other variables, we assume that a certain generalized coordinateqk and its derivative appear together as a single function
in the Hamiltonian
In that case, the function S can be partitioned into two functions, one that depends only on qk and another that depends only on the remaining generalized coordinates
Substitution of these formulae into the Hamilton-Jacobi equation shows that the function ? must be a constant (denoted here as ?k), yielding a first-order ordinary differential equation for Sk(qk).
In fortunate cases, the function S can be separated completely into N functions Sm(qm)
The separability of S depends both on the Hamiltonian and on the choice of generalized coordinates. For orthogonal coordinates and Hamiltonians that have no time dependence and are quadratic in the generalized momenta, S will be completely separable if the potential energy is additively separable in each coordinate, where the potential energy term for each coordinate is multiplied by the coordinate-dependent factor in the corresponding momentum term of the Hamiltonian (the Staeckel conditions). For illustration, several examples in orthogonal coordinates are worked in the next sections.
Examples in various coordinate systems
In spherical coordinates the Hamiltonian of a free particle moving in a conservative potential U can be written
The Hamilton-Jacobi equation is completely separable in these coordinates provided that there exist functions Ur(r), U?(?) and U?(?) such that U can be written in the analogous form
that, when solved, provide a complete solution for S.
Eikonal approximation and relationship to the Schrödinger equation
The isosurfaces of the function S(q; t) can be determined at any time t. The motion of an S-isosurface as a function of time is defined by the motions of the particles beginning at the points q on the isosurface. The motion of such an isosurface can be thought of as a wave moving through q space, although it does not obey the wave equation exactly. To show this, let S represent the phase of a wave
where ? is a constant (Planck's constant) introduced to make the exponential argument dimensionless; changes in the amplitude of the wave can be represented by having S be a complex number. We may then rewrite the Hamilton-Jacobi equation as
implying the particle moving along a circular trajectory with a permanent radius and an invariable value of momentum directed along a magnetic field vector.
b) For the flat, monochromatic, linearly polarized wave with a field directed along the axis
implying the particle figure-8 trajectory with a long its axis oriented along the electric field vector.
c) For the electromagnetic wave with axial (solenoidal) magnetic field:
where is the magnetic field magnitude in a solenoid with the effective radius , inductivity , number of windings , and an electric current magnitude through the solenoid windings. The particle motion occurs along the figure-8 trajectory in plane set perpendicular to the solenoid axis with arbitrary azimuth angle due to axial symmetry of the solenoidal magnetic field.
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