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Hamilton-Jacobi equations

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In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a type-2 canonical transformation of the classical Hamiltonian [H(q_,\dots,q_;p_,\dots,p_;t)] that results in a first-order, non-linear differential equation

[H\left(q_,\dots,q_;\frac},\dots,\frac};t\right) + \frac=0]
whose solution [S(q_,\dots,q_; t)] describes the behavior of the system. The HJE is a single differential equation of the [N] generalized coordinates; for comparison, Hamilton's equations of motion are a system of [2N] first-order equations. The function [S] is called Hamilton's principal function and, remarkably, equals the classical action. The Hamilton-Jacobi equations were developed initially for solving classical mechanics problems, and are excellent at identifying constants of motion.

The HJE are yet another equivalent expression of Hamilton's principle, similar to Hamilton's equations and the Euler-Lagrange equations. Hence, 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.

Contents

Notation

For brevity, we use boldface variables such as [\mathbf] to represent the list of [N] generalized coordinates

[\mathbf \equiv (q_, q_, \ldots, q_, q_)]
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 \cdot \mathbf \equiv \sum_^ p_ q_]

Derivation

Any canonical transformation involving a type-2 generating function [G_(\mathbf,\mathbf,t)] leads to the relations

[\qquad \over \partial \mathbf} = \mathbf, \qquad \over \partial \mathbf} = \mathbf, \qquadK = H + \over \partial t}]
(See the canonical transformation article for more details.)

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

[ \over dt} = \over dt} = 0]
i.e., the new generalized coordinates and momenta are constants of motion. The new generalized momenta [\mathbf] are usually denoted [\alpha_, \alpha_, \ldots, \alpha_, \alpha_], i.e., [P_ = \alpha_].

The HJE results from the equation for the transformed Hamiltonian [K]

[K(\mathbf,\mathbf,t) = H(\mathbf,\mathbf,t) + = 0.]
which is equivalent to the HJE

[H\left(\mathbf,},t\right) + = 0,]
since [\mathbf=\partial S/\partial \mathbf].

The new generalized coordinates [\mathbf] are also constants, typically denoted as [\beta_, \beta_, \ldots, \beta_, \beta_]. Once we have solved for [S(\mathbf,\boldsymbol\alpha, t)], these also give useful equations

[\mathbf = \boldsymbol\beta = ]
or written in components for clarity

[Q_ = \beta_ = \frac,\boldsymbol\alpha, t)}}]
Ideally, these [N] equations can be inverted to find the original generalized coordinates [\mathbf] as a function of the constants [\boldsymbol\alpha] and [\boldsymbol\beta], 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] in the HJE must be a constant (usually denoted [-E]), giving the separated solution

[S = W(q_,\dots,q_) - Et]
where the time-independent function [W(\mathbf)] is sometimes called Hamilton's characteristic function. The reduced Hamilton-Jacobi equation can then be written

[H\left(\mathbf,\frac} \right) = E]
To illustrate separability for other variables, we assume that a certain generalized coordinate [q_] and its derivative [\frac}] appear together in the Hamiltonian as a single function [\psi \left(q_, \frac} \right)]

[H = H(q_,\dots,q_, q_, \ldots, q_;p_, \dots, p_, p_, \ldots, p_; \psi; t)]
In that case, the function [S] can be partitioned into two functions, one that depends only on [q_] and another that depends only on the remaining generalized coordinates

[S = S_(q_) + S_(q_, \dots, q_, q_, \ldots, q_; t)]
Substitution of these formulae into the Hamilton-Jacobi equation shows that the function [\psi] must be a constant (denoted here as [\Gamma_]), yielding a first-order ordinary differential equation for [S_(q_)]

[\psi \left(q_, \frac}} \right) = \Gamma_]
In fortunate cases, the function [S] can be separated completely into [N] functions [S_(q_)]

[S=S_(q_)+S_(q_)+\cdots+S_(q_)-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 \left[ p_^ + frac^}} + frac^} sin^ 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) + \frac(\theta)}} + \frac(\phi)}\sin^\theta}]
where [U_(r)], [U_(\theta)] and [U_(\phi)] are arbitrary functions. Substitution of the completely separated solution [S = S_(r) + S_(\theta) + S_(\phi) - Et] into the HJE yields

[\frac \left( \frac} \right)^ + U_(r) + \frac} \left[ left( frac} right)^ + 2m U_(theta) right] + \frac\sin^\theta} \left[ left( frac} right)^ + 2m U_(phi) right] = E]
This equation may be solved by successive integrations of ordinary differential equations, beginning with the [\phi] equation

[\left( \frac} \right)^ + 2m U_(\phi) = \Gamma_]
where [\Gamma_] is a constant of the motion that eliminates the [\phi] dependence from the Hamilton-Jacobi equation

[\frac \left( \frac} \right)^ + U_(r) + \frac} \left[ left( frac} right)^ + 2m U_(theta) + frac}theta} right] = E]
The next ordinary differential equation involves the [\theta] generalized coordinate

[\left( \frac} \right)^ + 2m U_(\theta) + \frac}\theta} = \Gamma_]
where [\Gamma_] is again a constant of the motion that eliminates the [\theta] dependence and reduces the HJE to the final ordinary differential equation

[\frac \left( \frac} \right)^ + U_(r) + \frac}} = 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_^} \left( \sinh^ \mu + \sin^ \nu\right)} + \frac^} + 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(\mu) + U_(\nu)} \mu + \sin^ \nu} + U_(z)]
where [U_(\mu)], [U_(\nu)] and [U_(z)] are arbitrary functions. Substitution of the completely separated solution [S = S_(\mu) + S_(\nu) + S_(z) - Et] into the HJE yields

[\frac \left( \frac} \right)^ + U_(z) + \frac \left( \sinh^ \mu + \sin^ \nu\right)} \left[ left( frac} right)^ + left( frac} right)^ + 2m a^ U_(mu) + 2m a^ U_(nu)right] = E]
Separating the first ordinary differential equation

[\frac \left( \frac} \right)^ + U_(z) = \Gamma_]
yields the reduced Hamilton-Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)

[\left( \frac} \right)^ + \left( \frac} \right)^ + 2m a^ U_(\mu) + 2m a^ U_(\nu) = 2ma^ \left( \sinh^ \mu + \sin^ \nu\right) \left( E - \Gamma_ \right)]
which itself may be separated into two independent ordinary differential equations

[\left( \frac} \right)^ + 2m a^ U_(\mu) + 2ma^ \left(\Gamma_ - E \right) \sinh^ \mu = \Gamma_]
[\left( \frac} \right)^ + 2m a^ U_(\nu) + 2ma^ \left(\Gamma_ - E \right) \sin^ \nu = \Gamma_]
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_^} + \tau^\right)} + \frac^} + 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(\sigma) + U_(\tau)} + \tau^} + U_(z)]
where [U_(\sigma)], [U_(\tau)] and [U_(z)] are arbitrary functions. Substitution of the completely separated solution [S = S_(\sigma) + S_(\tau) + S_(z) - Et] into the HJE yields

[\frac \left( \frac} \right)^ + U_(z) + \frac + \tau^ \right)} \left[ left( frac} right)^ + left( frac} right)^ + 2m U_(sigma) + 2m U_(tau)right] = E]
Separating the first ordinary differential equation

[\frac \left( \frac} \right)^ + U_(z) = \Gamma_]
yields the reduced Hamilton-Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)

[\left( \frac} \right)^ + \left( \frac} \right)^ + 2m U_(\sigma) + 2m U_(\tau) = 2m \left( \sigma^ + \tau^ \right) \left( E - \Gamma_ \right)]
which itself may be separated into two independent ordinary differential equations

[\left( \frac} \right)^ + 2m U_(\sigma) + 2m\sigma^ \left(\Gamma_ - E \right) = \Gamma_]
[\left( \frac} \right)^ + 2m a^ U_(\tau) + 2m \tau^ \left(\Gamma_ - E \right) = \Gamma_]
that, when solved, provide a complete solution for [S].

Eikonal approximation and relationship to the Schrödinger equation

The isosurfaces of the function [S(\mathbf; 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] on the isosurface. The motion of such an isosurface can be thought of as a wave moving through [\mathbf] space, although it does not obey the wave equation exactly. To show this, let [S] represent the phase of a wave

[\psi = \psi_ e^]
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} \left( \boldsymbol\nabla \psi \right)^ - U\psi = \frac \frac]
which is a nonlinear variant of the Schrödinger equation. Conversely, starting with the Schrödinger equation and our Ansatz for [\psi], we arrive at a variant of the Hamilton-Jacobi equation

[\frac \left( \boldsymbol\nabla S \right)^ + U + \frac = \frac \nabla^ S]
Thus, the Hamilton-Jacobi equation is the classical limit ([\hbar \rightarrow 0]) of the Schrödinger equation.

Geodesics as Hamiltonian flows

It is frequently said that geodesics are "straight lines in curved space". By using the Hamilton-Jacobi approach to the geodesic equation, this statement can be given a very intuitive meaning: geodesics describe the motions of particles that are not experiencing any forces. In flat space, it is well known that a particle moving in a straight line will continue to move in a straight line if it experiences no external forces; this is Newton's first law. The Hamiltonan describing such motion is well known to be [H=mv^2/2=p^2/2m] with p being the momentum. It is the conservation of momentum that leads to the straight motion of a particle. On a curved surface, exactly the same ideas are at play, except that, in order to measure distances correctly, one must use the metric. To measure momenta correctly, one must use the inverse of the metric. The motion of a free particle on a curved surface still has exactly the same form as above, i.e. consisting entirely of a kinetic term. The resulting motion is still, in a sense, a "straight line", which is why it is sometimes said that geodesics are "straight lines in curved space". This idea is developed in greater detail below.

Geodesics as an application of the principle of least action

Given a (pseudo-)Riemannian manifold M, a geodesic may be defined as the curve that results from the application of the principle of least action. A differential equation describing their shape may be derived, using variational principles, by minimizing (or finding the extremum) of the energy of a curve. Given a smooth curve 
[\gamma:I\to M]



that maps an interval I of the real number line to the manifold M, one writes the energy



[E(\gamma)=\frac\int_I g(\dot\gamma(t),\dot\gamma(t))\,dt,]



where [\dot\gamma(t)] is the tangent vector to the curve [\gamma] at point [t \in I].
Here, [g(\cdot,\cdot)] is the metric tensor on the manifold M. 

Using the energy given above as the action, one may choose to solve either the Euler-Lagrange equations, or the Hamilton-Jacobi equations. Both methods give the geodesic equation as the solution; however, the Hamilton-Jacobi equations provide greater insight into the structure of the manifold, as shown below. In terms of the local coordinates on M, the (Euler-Lagrange) geodesic equation is  



[\frac + \Gamma^ _\frac\frac = 0]



Here, the xa(t) are the coordinates of the curve γ(t) and [\Gamma^ _] are the Christoffel symbols.  Repeated indecies imply the use of the summation convention.



Hamiltonian approach to the geodesic equations

Geodesics can be understood to be the Hamiltonian flows of a special Hamiltonian vector field defined on the cotangent space of the manifold. The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form consisting entirely of the kinetic term. 


The geodesic equations are second-order differential equations; they can be re-expressed as first-order ordinary differential equations taking the form of the Hamiltonian-Jacobi equations by introducing additional independent variables, as shown below.  Start by finding a chart that trivializes the cotangent bundle T∗M (i.e. a local trivialization):



[T^*M|_\simeq U \times \mathbb^n]



where U is an open subset of the manifold M, and the tangent space is of rank n.  Label the coordinates of the chart as (x1, x2, …, xn, p1, p2, …, pn). Then introduce the Hamiltonian as



[H(x,p)=\fracg^(x)p_a p_b.]



Here, gab(x) is the inverse of the metric tensor: gab(x)gbc(x) = [\delta^a_c].  This inverse almost always exists for a broad class of metric manifolds. The behaviour of the metric tensor under coordinate transformations implies that H is invariant under a change of variable. The geodesic equations can then be written as



[\dot^a = \frac = g^(x) p_b]



and



[\dot_a = - \frac  = -\frac \frac (x)} p_b p_c.]



The second order geodesic equations are easily obtained by substitution of one into the other.  The flow determined by these equations is called the cogeodesic flow. The first of the two equations gives the flow on the tangent bundle TM, the geodesic flow.  Thus, the geodesic lines are the integral curves of the geodesic flow onto the manifold M.  This is a Hamiltonian flow, and that the Hamiltonian is constant along the geodesics:



[\frac = \frac  \dot^a +\frac \dot_a = - \dot_a \dot^a + \dot^a \dot_a = 0.]


Thus, the geodesic flow splits the cotangent bundle into level sets of constant energy 



[M_E = \]



for each energy E ≥ 0, so that 



[T^*M=\bigcup_ M_E].



The Hopf-Rinow theorem guarantees the completeness of the manifold. The positivity of the energy follows from the positivity of the metric tensor; this analysis is modified on pseudo-Riemannian manifolds.



References for this section



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