Hamilton–Jacobi equation


In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.
The Hamilton–Jacobi equation is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, it fulfilled a long-held goal of theoretical physics 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 Hamilton–Jacobi equation is considered the "closest approach" of classical mechanics to quantum mechanics.
In mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. It can be understood as a special case of the Hamilton–Jacobi–Bellman equation from dynamic programming.

Notation

Boldface variables such as represent a list of generalized coordinates,
A dot over a variable or list signifies the time derivative. For example,
The dot product notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, such as

Mathematical formulation

Given the Hamiltonian of a mechanical system the Hamilton–Jacobi equation is written as a first-order, non-linear partial differential equation for the Hamilton's principal function,
The Hamilton's principal function is defined as the function of the upper limit of the action integral taken along the minimal action trajectory of the system,
where is the Lagrangian of the system and where the trajectory satisfies the Euler–Lagrange equation of the system,
Calculating the variation of with respect to variation of the end-point coordinate,
leads to
Using this result and calculating the variation of with respect to variation of the time of the end-point leads directly to the Hamilton–Jacobi equation,
or
where is the change of the trajectory at the old end-point due to the time shift and where is the Hamiltonian of the system.
Alternatively, as described below, the Hamilton–Jacobi equation may be derived from Hamiltonian mechanics by treating as the generating function for a canonical transformation of the classical Hamiltonian
The conjugate momenta correspond to the first derivatives of with respect to the generalized coordinates
As a solution to the Hamilton–Jacobi equation, the principal function contains undetermined constants, the first of them denoted as, and the last one coming from the integration of.
The relationship between and then describes the orbit in phase space in terms of these constants of motion. Furthermore, the quantities
are also constants of motion, and these equations can be inverted to find 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 of the generalized coordinates and the time. The generalized momenta do not appear, except as derivatives of. Remarkably, the function 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, 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.
Since the HJE is an equivalent expression of an integral minimization problem such as Hamilton's principle, the HJE can be useful in other problems of the calculus of variations and, more generally, in other branches of mathematics and physics, such as dynamical systems, symplectic geometry and quantum chaos. For example, the Hamilton–Jacobi equations can be used to determine the geodesics on a Riemannian manifold, an important variational problem in Riemannian geometry.

Derivation using a canonical transformation

Any canonical transformation involving a type-2 generating function leads to the relations
and Hamilton's equations in terms of the new variables and new Hamiltonian have the same form:
To derive the HJE, we choose a generating function in such a way that, it will make the new Hamiltonian. 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 are usually denoted, i.e. and the new generalized coordinates are typically denoted as, so.
Setting the generating function equal to Hamilton's principal function, plus an arbitrary constant :
the HJE automatically arises
Once we have solved for, 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 coordinates as a function of the constants and, 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 is:
so the time derivative of S is
Therefore,
so S is actually the classical action plus an undetermined constant.
When H does not explicitly depend on time,
in this case W is the same as abbreviated action.

Separation of variables

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, giving the separated solution
where the time-independent function 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 coordinate 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 ψ must be a constant, yielding a first-order ordinary differential equation for
In fortunate cases, the function can be separated completely into functions
In such a case, the problem devolves to ordinary differential equations.
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, 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. For illustration, several examples in orthogonal coordinates are worked in the next sections.

Examples in various coordinate systems

Spherical coordinates

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, Uθ and Uϕ such that U can be written in the analogous form
Substitution of the completely separated solution
into the HJE yields
This equation may be solved by successive integrations of ordinary differential equations, beginning with the equation for
where is a constant of the motion that eliminates the dependence from the Hamilton–Jacobi equation
The next ordinary differential equation involves the generalized coordinate
where is again a constant of the motion that eliminates the dependence and reduces the HJE to the final ordinary differential equation
whose integration completes the solution for.

Elliptic cylindrical coordinates

The Hamiltonian in elliptic cylindrical coordinates can be written
where the foci of the ellipses are located at on the -axis. The Hamilton–Jacobi equation is completely separable in these coordinates provided that has an analogous form
where :, and are arbitrary functions. Substitution of the completely separated solution
Separating the first ordinary differential equation
yields the reduced Hamilton–Jacobi equation
which itself may be separated into two independent ordinary differential equations
that, when solved, provide a complete solution for.

Parabolic cylindrical coordinates

The Hamiltonian in parabolic cylindrical coordinates can be written
The Hamilton–Jacobi equation is completely separable in these coordinates provided that has an analogous form
where,, and are arbitrary functions. Substitution of the completely separated solution
into the HJE yields
Separating the first ordinary differential equation
yields the reduced Hamilton–Jacobi equation
which itself may be separated into two independent ordinary differential equations
that, when solved, provide a complete solution for.

Waves and particles

Optical wave fronts and trajectories

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 medium's index of refraction 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 equationfor, where is the Lagrangian. The trajectories are then recovered from the knowledge of.

Relationship to the Schrödinger equation

The isosurfaces of the function can be determined at any time t. The motion of an -isosurface as a function of time is defined by the motions of the particles beginning at the points on the isosurface. The motion of such an isosurface can be thought of as a wave moving through -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 introduced to make the exponential argument dimensionless; changes in the amplitude of the wave can be represented by having be a complex number. We may then rewrite the Hamilton–Jacobi equation as
which is the Schrödinger equation.
Conversely, starting with the Schrödinger equation and our ansatz for, we arrive at
The classical limit of the Schrödinger equation above becomes identical to the following variant of the Hamilton–Jacobi equation,

Applications

HJE in a gravitational field

Using the energy–momentum relation in the form
for a particle of rest mass travelling in curved space, where are the contravariant coordinates of the metric tensor solved from the Einstein field equations, and is the speed of light. Setting the four-momentum equal to the four-gradient of the action,
gives the Hamilton–Jacobi equation in the geometry determined by the metric :
in other words, in a gravitational field.

HJE in electromagnetic fields

For a particle of rest mass and electric charge moving in electromagnetic field with four-potential in vacuum, the Hamilton–Jacobi equation in geometry determined by the metric tensor has a form
and can be solved for the Hamilton principal action function to obtain further solution for the particle trajectory and momentum:
where and with the cycle average of the vector potential.

A circularly polarized wave

In the case of circular polarization,
Hence
where,
implying the particle moving along a circular trajectory with a permanent radius and an invariable value of momentum directed along a magnetic field vector.

A monochromatic linearly polarized plane wave

For the flat, monochromatic, linearly polarized wave with a field directed along the axis
hence
implying the particle figure-8 trajectory with a long its axis oriented along the electric field vector.

An electromagnetic wave with a solenoidal magnetic field

For the electromagnetic wave with axial magnetic field:
hence
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.