# Hamilton–Jacobi equation

Template:Classical mechanics
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, it is a 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 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 that need not transform like a vector under rotation, e.g.,

A dot over a variable or list signifies the time derivative, 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.,

The dot product (also known as an "inner product") maps the two coordinate lists into one variable representing a single numerical value.

## Mathematical formulation

The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation^{[3]}

where

is the **classical Hamiltonian function**,

is called **Hamilton's principal function** (also the action, see below), *q _{i}* are the

*N*generalized coordinates (

*i*= 1,2...

*N*) which define the configuration of the system, and

*t*is time.

As described below, this 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 *α*_{1}, *α*_{2} ... *α _{N}*, 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 *q*_{1}...*q _{N}* 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 2*N* first-order equations for the time evolution of the generalized coordinates and their conjugate momenta *p*_{1}...*p _{N}*.

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

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Any canonical transformation involving a type-2 generating function *G*_{2}(**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 *G*_{2}(**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 *α*_{1}, *α*_{2} ... *α _{N}*, i.e.

*P*=

_{m}*α*, and the new generalized coordinates

_{m}**Q**are typically denoted as

*β*

_{1},

*β*

_{2}...

*β*, so

_{N}*Q*=

_{m}*β*.

_{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**, **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 *q _{k}* 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 *q _{k}* 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 Γ_{k}), yielding a first-order ordinary differential equation for *S _{k}(q_{k})*

In fortunate cases, the function *S* can be separated completely into *N* functions *S _{m}*(

*q*)

_{m}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 *U _{r}*(

*r*),

*U*(

_{θ}*θ*) and

*U*(

_{ϕ}*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 Γ_{ϕ} 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 Γ_{θ} 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 ±*a* on the *x*-axis. The Hamilton–Jacobi equation is completely separable in these coordinates provided that *U* has an analogous form

where *U _{μ}*(

*μ*),

*U*(

_{η}*η*) and

*U*(

_{z}*z*) are arbitrary functions. Substitution of the completely separated solution

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 *U _{σ}*(

*σ*),

*U*(

_{τ}*τ*) and

*U*(

_{z}*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

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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 unitless; 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 a *nonlinear* variant of the Schrödinger equation.

Conversely, starting with the Schrödinger equation and our ansatz for *ψ*, we arrive at
^{[5]}

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;^{[6]}

for a particle of rest mass *m* travelling in curved space, where *g ^{αβ}* 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

*P*equal to the four-gradient of the action

_{α}*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:^{[7]}

where and averaging procedure was performed to reveal the particle closed periodic motion. 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

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:^{[8]}

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.

## See also

- Canonical transformation
- Constant of motion
- Hamiltonian vector field
- Hamilton–Jacobi–Bellman equation in control theory
- Hamilton–Jacobi–Einstein equation
- WKB approximation
- William Rowan Hamilton
- Carl Gustav Jacob Jacobi
- Action-angle coordinates

## References

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- ↑ Sakurai, pp. 103–107.
- ↑ L.N. Hand, J.D. Finch. Analytical Mechanics, Cambridge University Press, 2008, ISBN 978-0-521-57572-0
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## Further reading

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