Jacobi equation

Hamilton–Jacobi equation

In physics, the Hamilton–Jacobi equation (HJE) is a reformulation of classical mechanics and, thus, 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.

Mathematical formulation

The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation for a function $S(q_{1},dots,q_{N}; t)$ called Hamilton's principal function

Hleft(q_{1},dots,q_{N};frac{partial S}{partial q_{1}},dots,frac{partial S}{partial q_{N}};tright) + frac{partial S}{partial t}=0.

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 $H(q_{1},dots,q_{N};p_{1},dots,p_{N};t)$ . The conjugate momenta correspond to the first derivatives of $S$ with respect to the generalized coordinates

p_{k} = frac{partial S}{partial q_{k}}.

which can be obtained as follows.
The change in the action from one path to a neighboring path is given by

delta S=sum_{i=1}^Nleft[frac{partial L}{partial dot{q}_{k}}delta q_kright]_{t_1}^{t^2}+sum_{i=1}^Nint_{t_1}^{t^2}left(frac {partial L}{partial q_k} - frac {d}{d t} frac {partial L}{partial dot{q}_k}right)delta q_k ,dt.

Since the paths of actual motion satisfy Euler-Lagrange equation, the integral in $delta S$ is zero. In the first term we put $delta q_k(t_1)=0$ , and denote the value of $delta q_k(t_2)$ by simply $delta q_k$ . Replacing $partial L/partial dot{q}_{k}$ by $p_k$ , we have finally

delta S=sum_{i=1}^N p_k delta q_k

From this relation it follows that the partial derivatives of the action with respect to the coordinates are equal to the corresponding momenta.
Q.E.D..
Similarly, the generalized coordinates can be obtained as derivatives with respect to the transformed momenta, as described below. By inverting these equations, one can determine the evolution of the mechanical system, i.e., determine the generalized coordinates as a function of time. The initial positions and velocities appear in the constants of integration for the solution

S

, which correspond to conserved quantities of the evolution such as the total energy, the angular momentum, or the Laplace-Runge-Lenz vector.

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 $q_{1},dots,q_{N}$ 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. As another comparison, Hamilton's equations of motion are likewise a system of $2N$ first-order equations for the time evolution of the generalized coordinates and their conjugate momenta $p_{1},dots,p_{N}$ .

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.

Notation

For brevity, we use boldface variables such as $mathbf{q}$ to represent the list of $N$ generalized coordinates

mathbf{q} stackrel{mathrm{def}}{=} (q_{1}, q_{2}, ldots, q_{N-1}, q_{N})

that need not transform like a vector under rotation. The dot product is defined here as the sum of the products of corresponding components, e.g.,

mathbf{p} cdot mathbf{q} stackrel{mathrm{def}}{=} sum_{k=1}^{N} p_{k} q_{k}.

Derivation

Any canonical transformation involving a type-2 generating function $G_{2}(mathbf{q},mathbf{P},t)$ leads to the relations

qquad {partial G_{2} over partial mathbf{q}} = mathbf{p}, qquad {partial G_{2} over partial mathbf{P}} = mathbf{Q}, qquad K = H + {partial G_{2} over partial t}

(See the canonical transformation article for more details.)

To derive the HJE, we choose a generating function $S(mathbf{q}, mathbf{P}, t)$ that makes the new Hamiltonian $K$ identically zero. Hence, all its derivatives are also zero, and Hamilton's equations become trivial

{dmathbf{P} over dt} = {dmathbf{Q} over dt} = 0

i.e., the new generalized coordinates and momenta are constants of motion. The new generalized momenta $mathbf{P}$ are usually denoted $alpha_{1}, alpha_{2}, ldots, alpha_{N-1}, alpha_{N}$ , i.e., $P_{m} = alpha_{m}$ .

The HJE results from the equation for the transformed Hamiltonian $K$

K(mathbf{Q},mathbf{P},t) = H(mathbf{q},mathbf{p},t) + {partial S over partial t} = 0.

which is equivalent to the HJE

Hleft(mathbf{q},{partial S over partial mathbf{q}},tright) + {partial S over partial t} = 0,

since $mathbf{p}=partial S/partial mathbf{q}$ .

The new generalized coordinates $mathbf{Q}$ are also constants, typically denoted as $beta_{1}, beta_{2}, ldots, beta_{N-1}, beta_{N}$ . Once we have solved for $S(mathbf{q},boldsymbolalpha, t)$ , these also give useful equations

mathbf{Q} = boldsymbolbeta = {partial S over partial boldsymbolalpha}

or written in components for clarity

Q_{m} = beta_{m} = frac{partial S(mathbf{q},boldsymbolalpha, t)}{partial alpha_{m}}

Ideally, these $N$ equations can be inverted to find the original generalized coordinates $mathbf{q}$ as a function of the constants $boldsymbolalpha$ and $boldsymbolbeta$ , thus solving the original problem.

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 $frac{partial S}{partial t}$ in the HJE must be a constant (usually denoted $-E$ ), giving the separated solution

S = W(q_{1},dots,q_{N}) - Et where the time-independent function

W(mathbf{q})

is sometimes called Hamilton's characteristic function. The reduced Hamilton–Jacobi equation can then be written

Hleft(mathbf{q},frac{partial S}{partial mathbf{q}} right) = E

To illustrate separability for other variables, we assume that a certain generalized coordinate $q_{k}$ and its derivative $frac{partial S}{partial q_{k}}$ appear together in the Hamiltonian as a single function $psi left(q_{k}, frac{partial S}{partial q_{k}} right)$

H = H(q_{1},dots,q_{k-1}, q_{k+1}, ldots, q_{N};p_{1}, dots, p_{k-1}, p_{k+1}, ldots, p_{N}; psi; t)

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

S = S_{k}(q_{k}) + S_{rem}(q_{1}, dots, q_{k-1}, q_{k+1}, ldots, q_{N}; t)

Substitution of these formulae into the Hamilton–Jacobi equation shows that the function $psi$ must be a constant (denoted here as $Gamma_{k}$ ), yielding a first-order ordinary differential equation for $S_{k}(q_{k})$

psi left(q_{k}, frac{d S_{k}}{d q_{k}} right) = Gamma_{k}

In fortunate cases, the function $S$ can be separated completely into $N$ functions $S_{m}(q_{m})$

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.

Example of spherical coordinates

The Hamiltonian in spherical coordinates 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 $U$ has an analogous form

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

where $U_{r}(r)$ , $U_{theta}(theta)$ and $U_{phi}(phi)$ are arbitrary functions. 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{dS_{r}}{dr} right)^{2} + U_{r}(r) + frac{1}{2m r^{2}} left[left(frac{dS_{theta}}{dtheta} right)^{2} + 2m U_{theta}(theta) right] + frac{1}{2m r^{2}sin^{2}theta} left[left(frac{dS_{phi}}{dphi} right)^{2} + 2m U_{phi}(phi) right] = E

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

left(frac{dS_{phi}}{dphi} right)^{2} + 2m U_{phi}(phi) = Gamma_{phi}

where $Gamma_{phi}$ is a constant of the motion that eliminates the $phi$ dependence from the Hamilton–Jacobi equation

frac{1}{2m} left(frac{dS_{r}}{dr} right)^{2} + U_{r}(r) + frac{1}{2m r^{2}} left[left(frac{dS_{theta}}{dtheta} right)^{2} + 2m U_{theta}(theta) + frac{Gamma_{phi}}{sin^{2}theta} right] = E

The next ordinary differential equation involves the $theta$ generalized coordinate

left(frac{dS_{theta}}{dtheta} right)^{2} + 2m U_{theta}(theta) + frac{Gamma_{phi}}{sin^{2}theta} = Gamma_{theta}

where $Gamma_{theta}$ is again a constant of the motion that eliminates the $theta$ dependence and reduces the HJE to the final ordinary differential equation

frac{1}{2m} left(frac{dS_{r}}{dr} right)^{2} + U_{r}(r) + frac{Gamma_{theta}}{2m r^{2}} = E

whose integration completes the solution for $S$ .

Example of elliptic cylindrical coordinates

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} nuright)} + frac{p_{z}^{2}}{2m} + U(mu, nu, z)

where the foci of the ellipses are located at $pm a$ 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 $U_{mu}(mu)$ , $U_{nu}(nu)$ and $U_{z}(z)$ 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{dS_{z}}{dz} right)^{2} + U_{z}(z) + frac{1}{2ma^{2} left(sinh^{2} mu + sin^{2} nuright)} left[left(frac{dS_{mu}}{dmu} right)^{2} + left(frac{dS_{nu}}{dnu} 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{dS_{z}}{dz} 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{dS_{mu}}{dmu} right)^{2} + left(frac{dS_{nu}}{dnu} right)^{2} + 2m a^{2} U_{mu}(mu) + 2m a^{2} U_{nu}(nu) = 2ma^{2} left(sinh^{2} mu + sin^{2} nuright) left(E - Gamma_{z} right)

which itself may be separated into two independent ordinary differential equations

left(frac{dS_{mu}}{dmu} right)^{2} + 2m a^{2} U_{mu}(mu) + 2ma^{2} left(Gamma_{z} - E right) sinh^{2} mu = Gamma_{mu}

left(frac{dS_{nu}}{dnu} 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 $S$ .

Example of parabolic cylindrical coordinates

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 $U_{sigma}(sigma)$ , $U_{tau}(tau)$ and $U_{z}(z)$ 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{dS_{z}}{dz} right)^{2} + U_{z}(z) + frac{1}{2m left(sigma^{2} + tau^{2} right)} left[left(frac{dS_{sigma}}{dsigma} right)^{2} + left(frac{dS_{tau}}{dtau} right)^{2} + 2m U_{sigma}(sigma) + 2m U_{tau}(tau)right] = E

Separating the first ordinary differential equation

frac{1}{2m} left(frac{dS_{z}}{dz} 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{dS_{sigma}}{dsigma} right)^{2} + left(frac{dS_{tau}}{dtau} 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{dS_{sigma}}{dsigma} right)^{2} + 2m U_{sigma}(sigma) + 2msigma^{2} left(Gamma_{z} - E right) = Gamma_{sigma}

left(frac{dS_{tau}}{dtau} right)^{2} + 2m a^{2} 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

The isosurfaces of the function $S(mathbf{q}; t)$ 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 $mathbf{q}$ on the isosurface. The motion of such an isosurface can be thought of as a wave moving through $mathbf{q}$ 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 $hbar$ is a 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 re-write the Hamilton–Jacobi equation as

frac{hbar^{2}}{2mpsi} left(boldsymbolnabla psi right)^{2} - Upsi = 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 $psi$ , we arrive at,

frac{1}{2m} left(boldsymbolnabla S right)^{2} + U + frac{partial S}{partial t} = frac{ihbar}{2m} nabla^{2} S

The classical limit ( $hbar rightarrow 0$ ) of the Schrödinger equation above becomes identical to the following variant of the Hamilton-Jacobi equation,

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

The Hamilton-Jacobi equation in the gravitational field

g^{ik}frac{partial{S}}{partial{x^{i}}}frac{partial{S}}{partial{x^{k}}} - m^{2}c^{2} = 0

where $g^{ik}$ are the contravariant components of the metric tensor, m is the rest mass of the particle and c is the speed of light.

References

Hamilton W. (1833) "On a General Method of Expressing the Paths of Light, and of the Planets, by the Coefficients of a Characteristic Function", Dublin University Review, pp. 795-826.

Hamilton W. (1834) "On the Application to Dynamics of a General Mathematical Method previously Applied to Optics", British Association Report, pp.513-518.