Hamilton–Jacobi equation

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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[edit]

Boldface variables such as \mathbf{q} represent a list of N generalized coordinates that need not transform like a vector under rotation, e.g.,


\mathbf{q} \equiv (q_{1}, q_{2}, \ldots, q_{N-1}, q_{N})

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


\dot{\mathbf{q}} \equiv \frac{d\mathbf{q}}{dt}
.

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.,


\mathbf{p} \cdot \mathbf{q} \equiv \sum_{k=1}^{N} p_{k} q_{k}.

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


Mathematical formulation[edit]

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

 H + \frac{\partial S}{\partial t}=0

where

H = H\left(q_1,\cdots,q_N;\frac{\partial S}{\partial q_1},\cdots,\frac{\partial S}{\partial q_N};t\right)\,\!

is the classical Hamiltonian function,

S = S(q_1,q_2\cdots q_N, t)

is called Hamilton's principal function (also the action, see below), qi 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

H = H(q_1,q_2\cdots q_N;p_1,p_2\cdots p_N;t).

The conjugate momenta correspond to the first derivatives of S with respect to the generalized coordinates

p_k = \frac{\partial S}{\partial q_k}.

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 \scriptstyle \partial S/\partial t .

The relationship between p and q then describes the orbit in phase space in terms of these constants of motion. Furthermore, the quantities

\beta_k=\frac{\partial S}{\partial\alpha_k},\quad k=1,2 \cdots N

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[edit]

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.

Derivation[edit]

See the canonical transformation article for more details.

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


\bold{p} = {\partial G_2 \over \partial \bold{q}}, \quad
\bold{Q} = {\partial G_2 \over \partial \bold{P}}, \quad
K(\bold{Q},\bold{P},t) = H(\bold{q},\bold{p},t) + {\partial G_2 \over \partial t}

and Hamilton's equations in terms of the new variables P, Q and new Hamiltonian K have the same form:

 \dot{\mathbf{P}}  = -{\partial K \over \partial \bold{Q}}, 
\quad \dot{\mathbf{Q}}  = +{\partial K \over \partial \bold{P}}.

To derive the HJE, we choose a generating function G2(q, P, t) that makes the new Hamiltonian K = 0. Hence, all its derivatives are also zero, and the transformed Hamilton's equations become trivial

\dot{\mathbf{P}} = \dot{\mathbf{Q}} = 0

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:

G_2(\bold{q},\boldsymbol{\alpha},t)=S(\bold{q},t)+A,

the HJE automatically arises:

\bold{p}=\frac{\partial G_2}{\partial \bold{q}}=\frac{\partial S}{\partial \bold{q}} \ \rightarrow \ 
H(\bold{q},\bold{p},t) + {\partial G_2 \over \partial t}=0 \ \rightarrow \ 
H\left(\bold{q},\frac{\partial S}{\partial \bold{q}},t\right) + {\partial S \over \partial t}=0.

Once we have solved for S(q, α, t), these also give us the useful equations


\bold{Q} = \boldsymbol\beta = 
{\partial S \over \partial \boldsymbol\alpha}

or written in components for clarity

 Q_{m} = \beta_{m} =  \frac{\partial S(\bold{q},\boldsymbol\alpha, t)}{\partial \alpha_{m}}.

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[edit]

Hamilton's principal function S and classical function H are both closely related to action. The total differential of S is:

 \mathrm{d}S =\sum_i \frac{\partial S}{\partial q_i} \mathrm{d}q_i + \frac{\partial S}{\partial t}\mathrm{d}t

so the time derivative of S is

\frac{\mathrm{d}S}{\mathrm{d}t} =\sum_i\frac{\partial S}{\partial q_i}\dot{q}_i+\frac{\partial S}{\partial t} =\sum_ip_i\dot{q}_i-H = L.

Therefore

S=\int L\,\mathrm{d}t ,

so S is actually the classical action plus an undetermined constant.

When H does not explicitly depend on time,

W=S+Et=S+Ht=\int(L+H)\,\mathrm{d}t=\int\bold{p}\cdot\mathrm{d}\bold{q},

in this case W is the same as abbreviated action.

Separation of variables[edit]

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 \scriptstyle \partial S/\partial t in the HJE must be a constant, usually denoted (–E), giving the separated solution

 S = W(q_1,q_2 \cdots q_N) - Et

where the time-independent function W(q) is sometimes called Hamilton's characteristic function. The reduced Hamilton–Jacobi equation can then be written

 H\left(\bold{q},\frac{\partial S}{\partial \bold{q}} \right) = E.

To illustrate separability for other variables, we assume that a certain generalized coordinate qk and its derivative \scriptstyle \partial S/\partial q_k appear together as a single function

\psi \left(q_k, \frac{\partial S}{\partial q_k} \right)

in the Hamiltonian

 H = H(q_1,q_2\cdots q_{k-1}, q_{k+1}\cdots q_N; p_1,p_2\cdots p_{k-1}, p_{k+1}\cdots p_N; \psi; t).

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

S = S_k(q_k) + S_{rem}(q_1\cdots q_{k-1}, q_{k+1} \cdots q_N, t).

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)

 \psi \left(q_k, \frac{\mathrm{d} S_k}{\mathrm{d} q_k} \right) = \Gamma_k.

In fortunate cases, the function S can be separated completely into N functions Sm(qm)

 S=S_1(q_1)+S_2(q_2)+\cdots+S_N(q_N)-Et.

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[edit]

Spherical coordinates[edit]

In spherical coordinates the Hamiltonian of a free particle moving in a conservative potential U can be written

 H = \frac{1}{2m} \left[ p_{r}^{2} + \frac{p_{\theta}^{2}}{r^{2}} + \frac{p_{\phi}^{2}}{r^{2} \sin^{2} \theta} \right] + U(r, \theta, \phi).

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

 U(r, \theta, \phi) = U_{r}(r) + \frac{U_{\theta}(\theta)}{r^{2}} + \frac{U_{\phi}(\phi)}{r^{2}\sin^{2}\theta} .

Substitution of the completely separated solution

S = S_{r}(r) + S_{\theta}(\theta) + S_{\phi}(\phi) - Et

into the HJE yields


\frac{1}{2m} \left( \frac{\mathrm{d}S_{r}}{\mathrm{d}r} \right)^{2} + U_{r}(r) + 
\frac{1}{2m r^{2}} \left[ \left( \frac{\mathrm{d}S_{\theta}}{\mathrm{d}\theta} \right)^{2} + 2m U_{\theta}(\theta) \right] + 
\frac{1}{2m r^{2}\sin^{2}\theta} \left[ \left( \frac{\mathrm{d}S_{\phi}}{\mathrm{d}\phi} \right)^{2} + 2m U_{\phi}(\phi) \right]  = E.

This equation may be solved by successive integrations of ordinary differential equations, beginning with the equation for ϕ

 \left( \frac{\mathrm{d}S_{\phi}}{\mathrm{d}\phi} \right)^{2} + 2m U_{\phi}(\phi) = \Gamma_{\phi}

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

 \frac{1}{2m} \left( \frac{\mathrm{d}S_{r}}{\mathrm{d}r} \right)^{2} + U_{r}(r) + \frac{1}{2m r^{2}} \left[ \left( \frac{\mathrm{d}S_{\theta}}{\mathrm{d}\theta} \right)^{2} + 2m U_{\theta}(\theta) + \frac{\Gamma_{\phi}}{\sin^{2}\theta} \right] = E.

The next ordinary differential equation involves the θ generalized coordinate

 \left( \frac{\mathrm{d}S_{\theta}}{\mathrm{d}\theta} \right)^{2} + 2m U_{\theta}(\theta) + \frac{\Gamma_{\phi}}{\sin^{2}\theta} = \Gamma_{\theta}

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

 \frac{1}{2m} \left( \frac{\mathrm{d}S_{r}}{\mathrm{d}r} \right)^{2} + U_{r}(r) + \frac{\Gamma_{\theta}}{2m r^{2}} = E

whose integration completes the solution for S.

Elliptic cylindrical coordinates[edit]

The Hamiltonian in elliptic cylindrical coordinates can be written

 H = \frac{p_{\mu}^{2} + p_{\nu}^{2}}{2ma^{2} \left( \sinh^{2} \mu + \sin^{2} \nu\right)} +  \frac{p_{z}^{2}}{2m}  + U(\mu, \nu, z)

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

 U(\mu, \nu, z) = \frac{U_{\mu}(\mu) + U_{\nu}(\nu)}{\sinh^{2} \mu + \sin^{2} \nu} + U_{z}(z)

where Uμ(μ), Uη(η) and Uz(z) are arbitrary functions. Substitution of the completely separated solution

S = S_{\mu}(\mu) + S_{\nu}(\nu) + S_{z}(z) - Et into the HJE yields

\frac{1}{2m} \left( \frac{\mathrm{d}S_{z}}{\mathrm{d}z} \right)^{2} + U_{z}(z) + 
\frac{1}{2ma^{2} \left( \sinh^{2} \mu + \sin^{2} \nu\right)} \left[ \left( \frac{\mathrm{d}S_{\mu}}{\mathrm{d}\mu} \right)^{2} + \left( \frac{\mathrm{d}S_{\nu}}{\mathrm{d}\nu} \right)^{2} + 2m a^{2} U_{\mu}(\mu) + 2m a^{2} U_{\nu}(\nu)\right] = E.

Separating the first ordinary differential equation

 \frac{1}{2m} \left( \frac{\mathrm{d}S_{z}}{\mathrm{d}z} \right)^{2} + U_{z}(z) = \Gamma_{z}

yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)

 \left( \frac{\mathrm{d}S_{\mu}}{\mathrm{d}\mu} \right)^{2} + \left( \frac{\mathrm{d}S_{\nu}}{\mathrm{d}\nu} \right)^{2} + 2m a^{2} U_{\mu}(\mu) + 2m a^{2} U_{\nu}(\nu) = 2ma^{2} \left( \sinh^{2} \mu + \sin^{2} \nu\right) \left( E - \Gamma_{z} \right)

which itself may be separated into two independent ordinary differential equations

 \left( \frac{\mathrm{d}S_{\mu}}{\mathrm{d}\mu} \right)^{2} + 2m a^{2} U_{\mu}(\mu) + 2ma^{2} \left(\Gamma_{z} - E \right) \sinh^{2} \mu = \Gamma_{\mu}
 \left( \frac{\mathrm{d}S_{\nu}}{\mathrm{d}\nu} \right)^{2} + 2m a^{2} U_{\nu}(\nu) + 2ma^{2} \left(\Gamma_{z} - E \right) \sin^{2} \nu  = \Gamma_{\nu}

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

Parabolic cylindrical coordinates[edit]

The Hamiltonian in parabolic cylindrical coordinates can be written

 H = \frac{p_{\sigma}^{2} + p_{\tau}^{2}}{2m \left( \sigma^{2} + \tau^{2}\right)} + \frac{p_{z}^{2}}{2m}  + U(\sigma, \tau, z).

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

 U(\sigma, \tau, z) = \frac{U_{\sigma}(\sigma) + U_{\tau}(\tau)}{\sigma^{2} + \tau^{2}} + U_{z}(z)

where Uσ(σ), Uτ(τ) and Uz(z) are arbitrary functions. Substitution of the completely separated solution

S = S_{\sigma}(\sigma) + S_{\tau}(\tau) + S_{z}(z) - Et

into the HJE yields


\frac{1}{2m} \left( \frac{\mathrm{d}S_{z}}{\mathrm{d}z} \right)^{2} + U_{z}(z) + 
\frac{1}{2m \left( \sigma^{2} + \tau^{2} \right)} \left[ \left( \frac{\mathrm{d}S_{\sigma}}{\mathrm{d}\sigma} \right)^{2} + \left( \frac{\mathrm{d}S_{\tau}}{\mathrm{d}\tau} \right)^{2} + 2m U_{\sigma}(\sigma) + 2m U_{\tau}(\tau)\right] = E.

Separating the first ordinary differential equation


\frac{1}{2m} \left( \frac{\mathrm{d}S_{z}}{\mathrm{d}z} \right)^{2} + U_{z}(z) = \Gamma_{z}

yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)


\left( \frac{\mathrm{d}S_{\sigma}}{\mathrm{d}\sigma} \right)^{2} + \left( \frac{\mathrm{d}S_{\tau}}{\mathrm{d}\tau} \right)^{2} + 2m U_{\sigma}(\sigma) + 2m U_{\tau}(\tau) = 2m \left( \sigma^{2} + \tau^{2} \right) \left( E - \Gamma_{z} \right)

which itself may be separated into two independent ordinary differential equations


\left( \frac{\mathrm{d}S_{\sigma}}{\mathrm{d}\sigma} \right)^{2} + 2m U_{\sigma}(\sigma) + 2m\sigma^{2} \left(\Gamma_{z} - E \right) = \Gamma_{\sigma}

\left( \frac{\mathrm{d}S_{\tau}}{\mathrm{d}\tau} \right)^{2} + 2m U_{\tau}(\tau) + 2m \tau^{2} \left(\Gamma_{z} - E \right) = \Gamma_{\tau}

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

Eikonal approximation and relationship to the Schrödinger equation[edit]

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

 \psi = \psi_{0} e^{iS/\hbar}

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

 \frac{\hbar^{2}}{2m\psi} \left( \nabla \psi \right)^{2} - U\psi = \frac{\hbar}{i} \frac{\partial \psi}{\partial t}

which is a nonlinear variant of the Schrödinger equation.

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

 \frac{1}{2m} \left( \nabla S \right)^{2} + U + \frac{\partial S}{\partial t} = \frac{i\hbar}{2m} \nabla^{2} S.

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

 \frac{1}{2m} \left( \nabla S \right)^{2} + U + \frac{\partial S}{\partial t} = 0.

HJE in a gravitational field[edit]

Using the energy–momentum relation in the form;[6]

g^{\alpha\beta}P_\alpha P_\beta + (mc)^2 = 0 \,,

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;

P_\alpha = \frac{\partial S}{\partial x^\alpha}

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

g^{\alpha\beta}\frac{\partial S}{\partial x^\alpha}\frac{\partial S}{\partial x^\beta} + (mc)^2 = 0\,,

in other words, in a gravitational field.

See also[edit]

References[edit]

  1. ^ Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: Addison-Wesley. pp. 484–492. ISBN 0-201-02918-9.  (particularly the discussion beginning in the last paragraph of page 491)
  2. ^ Sakurai, pp. 103–107.
  3. ^ Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978-0-521-57572-0
  4. ^ Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: Addison-Wesley. p. 440. ISBN 0-201-02918-9. 
  5. ^ Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: Addison-Wesley. pp. 490–491. ISBN 0-201-02918-9. 
  6. ^ J.A. Wheeler, C. Misner, K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 649. ISBN 0-7167-0344-0. 

Further reading[edit]