Hamilton–Jacobi equation

In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

In physics, the Hamilton–Jacobi equation is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion[citation needed], 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 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.[1][2]

Notation

Boldface variables such as represent a list of generalized coordinates,

A dot over a variable or list signifies the time derivative (see Newton's notation), e.g.,

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, e.g.,

Mathematical formulation

Given the Hamiltonian of a mechanical system (where , are coordinates and momenta of the system and is time) the Hamilton–Jacobi equation is written as a first-order, non-linear partial differential equation for the Hamilton's principal function [3],

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,

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 S as the generating function for a canonical transformation of the classical Hamiltonian

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 α1, α2 ... αN, and the last one coming from the integration of .

The relationship between p and q 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 q as a function of all the α and β constants and time.[4]

Comparison with other formulations of mechanics

The HJE is a single, first-order partial differential equation for the function S of the N generalized coordinates q1...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.

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.

Wave-trajectory duality

The HJE establishes a duality between trajectories and wave fronts.[5] 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 .

Derivation using canonical transformation

Any canonical transformation involving a type-2 generating function G2(q, P, t) leads to the relations

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 α1, α2 ... αN, i.e. Pm = αm, and the new generalized coordinates Q are typically denoted as β1, β2 ... βN, so Qm = βm.

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, α, 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 coordinates q as a function of the constants α, β 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:

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 (–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 coordinate qk 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 (denoted here as Γk), 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)

In such a case, the problem devolves to N 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, 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

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(r), Uθ(θ) and Uϕ(ϕ) 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 ϕ

where Γϕ is a constant of the motion that eliminates the ϕ dependence from the Hamilton–Jacobi equation

The next ordinary differential equation involves the θ generalized coordinate

where Γθ is again a constant of the motion that eliminates the θ dependence and reduces the HJE to the final ordinary differential equation

whose integration completes the solution for S.

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 U 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 (after re-arrangement and multiplication of both sides by the denominator)

which itself may be separated into two independent ordinary differential equations

that, when solved, provide a complete solution for S.

Parabolic cylindrical coordinates

The Hamiltonian in parabolic cylindrical coordinates can be written

The Hamilton–Jacobi equation is completely separable in these coordinates provided that U has an analogous form

where Uσ(σ), Uτ(τ) and Uz(z) 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 (after re-arrangement and multiplication of both sides by the denominator)

which itself may be separated into two independent ordinary differential equations

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 ħ 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

which is the Schrödinger equation.

Conversely, starting with the Schrödinger equation and our ansatz for ψ, we arrive at [6]

The classical limit (ħ → 0) of the Schrödinger equation above becomes identical to the following variant of the Hamilton–Jacobi equation,

HJE in a gravitational field

Using the energy–momentum relation in the form;[7]

for a particle of rest mass m travelling in curved space, where gαβ are the contravariant coordinates of the metric tensor (i.e., the inverse metric) solved from the Einstein field equations, and c is the speed of light, setting the four-momentum Pα equal to the four-gradient of the action S;

gives the Hamilton–Jacobi equation in the geometry determined by the metric g:

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:[8]

,

where and with the cycle average of the vector potential. Therefore:

a) For a wave with the 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.

b) 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.

c) For the electromagnetic wave with axial (solenoidal) magnetic field:[9]

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.

References

1. ^ Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: Addison-Wesley. pp. 484–492. ISBN 978-0-201-02918-5. (particularly the discussion beginning in the last paragraph of page 491)
2. ^ Sakurai, pp. 103–107.
3. ^ Hand, L. N.; Finch, J. D. (2008). Analytical Mechanics. Cambridge University Press. ISBN 978-0-521-57572-0.
4. ^ Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: Addison-Wesley. p. 440. ISBN 978-0-201-02918-5.
5. ^ Houchmandzadeh, Bahram (2020). "The Hamilton-Jacobi Equation : an alternative approach". American Journal of Physics. 85 (5): 10.1119/10.0000781. doi:10.1119/10.0000781.
6. ^ Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: Addison-Wesley. pp. 490–491. ISBN 978-0-201-02918-5.
7. ^ J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 649, 1188. ISBN 978-0-7167-0344-0.
8. ^ L. Landau and E. Lifshitz. THE CLASSICAL THEORY OF FIELDS. ADDISON-WESLEY PUBLISHING COMPANY, INC., Reading, Massachusetts, USA 1959.
9. ^ E. V. Shun'ko; D. E. Stevenson; V. S. Belkin (2014). "Inductively Coupling Plasma Reactor With Plasma Electron Energy Controllable in the Range from ~6 to ~100 eV". IEEE Transactions on Plasma Science. 42, part II (3): 774–785. Bibcode:2014ITPS...42..774S. doi:10.1109/TPS.2014.2299954.