Hamilton-Jacobi equation

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Author: Dr. Robert L. Warnock, SLAC National Accelerator Laboratory and Lawrence Berkeley National Laboratory, CA

The Hamilton-Jacobi Equation is a first order nonlinear partial differential equation of the form $H(x,u_x(x,\alpha,t),t)+u_t(x,\alpha,t)=K(\alpha,t)$ with independent variables $(x,t)\in {\mathbb R}^n\times{\mathbb R}$ and parameters $\alpha\in {\mathbb R}^n$ . It has wide applications in optics, mechanics, and semi-classical quantum theory. Its solutions determine infinite families of solutions of Hamilton's ordinary differential equations, which are the equations of motion of a mechanical system or an optical system in the ray approximation.

1 History
2 Canonical Transformation
3 Hamilton-Jacobi Equation and Invariant Tori
4 Action as a Solution of the Hamilton-Jacobi Equation
5 Solution of classical problems by separation of variables
6 Applications and Numerical Methods
7 Bibliography

[edit]

History

Sir William Rowan Hamilton (1805-1865) carried out one of the earliest studies of geometrical optics in an arbitrary medium with varying index of refraction (Hamilton (1830-1832), Synge (1937), Carathéodory (1937)). He found an eloquent expression of the topic in a characteristic function, which is the optical path length of a ray, regarded as a function of initial and final positions and times of the ray. This and related functions satisfy partial differential equations, and directly determine infinite families of rays. Following an analogy between rays and trajectories of a mechanical system, Hamilton soon extended his concepts to mechanics, incorporating ideas of Lagrange and others concerning generalized coordinates. The resulting Hamiltonian mechanics, notable for its invariance under coordinate transformations, is a cornerstone of theoretical physics.

With an emphasis on mechanics, Carl Gustav Jacob Jacobi (1804-1851) sharpened Hamilton's formulation, clarified mathematical issues, and made significant applications (Jacobi (1842-1843)). The resultant Hamilton-Jacobi theory and later developments are presented in several famous texts: Arnol'd (1974), Landau & Lifshitz (1969), Gantmacher (1970), Born & Wolf (1965), Lanczos (1949), Carathéodory (1982), Courant & Hilbert (1962). For studies using modern PDE theory see Lions (1982), Evans (2008), and Benton (1977). The theory embodies a wave-particle duality, which figured in the advent of the de Broglie - Schrödinger wave mechanics (Butterfield (2005)).

In a view broader than that of the original work, a solution of the Hamilton-Jacobi equation is the generator of a canonical transformation, a symplectic change of variables intended to simplify the equations of motion. In this framework (as applied to mechanics) there are solutions of a type different from that of Hamilton, which determine not only orbits but also invariant tori in phase space on which the orbits lie. These solutions, which are known to exist only under special circumstances, are the subject of the celebrated work of Kolmogorov, Arnol'd, and Moser; see Gallavotti (1983). Even approximate invariants, constructed by approximate solutions of the Hamilton-Jacobi equation, have implications for stability of motion over finite times (Nekhoroshev (1977), Warnock & Ruth (1992)). Approximate invariants also find applications in the Einstein-Brillouin-Keller quantization of semi-classical quantum theory (Percival (1977), Chapman et al. (1976), Martens & Ezra (1987)). Various forms and generalizations of the Hamilton-Jacobi equation occur widely in contemporary applied mathematics, for instance in optimal control theory (Fleming & Rishel (1975)).

[edit]

Canonical Transformation

Canonical transformations (equivalently, symplectic transformations) are of crucial importance in classical mechanics, as they are the chief means of solving a mechanical system or clarifying the structure of the system when it cannot be solved. The Hamilton-Jacobi equation is used to generate particular canonical transformations that simplify the equations of motion.

A mechanical system with $n$ degrees of freedom is described by generalized coordinates $q=(q_1,\cdots, q_n)$ and corresponding generalized momenta $p=(p_1,\cdots,p_n)$ ; we write $z=(q,p)$ . The motion of the system is governed by Hamilton's canonical equations of motion, i.e., the ordinary differential equations

(1)

$\dot q= H_p(z,t)\ ,\quad \dot p=-H_q(z,t)\ ,$

where $\dot{}\$ denotes the time derivative and subscripts indicate vectors of partial derivatives; thus $H_q=(\partial H/\partial q_1,\cdots,\partial H/\partial q_n)$ . The Hamiltonian function $H:\mathbb{R}^{2n}\times\mathbb{R}\rightarrow\mathbb{R}$ is here assumed to be $C^2$ in $z$ and continuous in $t$ . The solution of the initial value problem for the Hamiltonian system (1) is denoted by ${\mathbf z}(t,z_0)=({\mathbf q}(t,z_0),{\mathbf p}(t,z_0))$ for initial value $z_0={\mathbf z}(0,z_0)$ . This solution, denoted by the bold faced letter $\mathbf z$ to distinguish it from a general point $z$ in phase space, will be called an orbit. If $H$ depends on the time, specification of an orbit requires the initial time $t_0$ (not just the elapsed time) as well as the initial condition $z_0$ ; for convenience the origin of time is chosen so that $t_0=0$ .

One seeks a transformation of coordinates, $Z=(Q,P)=\Phi(z,t)=(\Phi_1(z,t),\Phi_2(z,t))$ , so that the equations of motion retain their form but with a new Hamiltonian $K$ , namely

(2)

$\dot Q= K_P(Z,t)\ ,\quad \dot P=- K_Q(Z,t)\ .$

If $K$ can be made independent of $Q$ , then $\mathbf P$ is constant and the solution of (2) is given simply as

(3)

${\mathbf Q}(t,Z_0)=Q_0+\int_0^t K_P(P_0,\tau)d\tau\ ,\quad {\mathbf P}(t,Z_0)=P_0\ .$

The solution of (1) is retrieved by the inverse transformation $z=\Psi(Z,t) \equiv \Phi^{-1}(Z,t)$ .

Write ${\mathbf Z}(t,Z_0)=({\mathbf Q}(t,Z_0),{\mathbf P}(t,Z_0))=\Phi({\mathbf z}(t,z_0),t)$ for an orbit in the new coordinates, where $Z_0=\Phi(z_0,0)$ . Reference to initial conditions will often be suppressed. A canonical transformation will be determined through the equation

(4)

${\mathbf p}(t)\cdot\dot{\mathbf q}(t)-H({\mathbf z}(t),t)={\mathbf P}(t)\cdot\dot{\mathbf Q}(t)-K({\mathbf Z}(t),t) +\frac{d}{dt}F_1({\mathbf q}(t),{\mathbf Q}(t),t)\ ,$

where $\cdot$ indicates the scalar product and the given function $F_1(q,Q,t)$ is $C^2$ in its first two arguments, $C^1$ in $t$ , and such that

(5)

$\det F_{1qQ}=\det\{\partial^2 F_1/\partial q_i\partial Qj\}\ne 0\ ,$

in some open region $\Omega\subset \mathbb{R}^{2n+1}$ of $(q,Q,t)$ -space. This function $F_1$ is called the generator or generating function of the transformation. By writing out $dF_1/dt$ , one sees that (4) is satisfied if

(6)

${\mathbf p}(t)=F_{1q}({\mathbf q}(t),{\mathbf Q}(t),t)\ ,$

(7)

${\mathbf P}(t)=-F_{1Q}({\mathbf q}(t),{\mathbf Q}(t),t)\ ,$

(8)

$K({\mathbf Z}(t),t)=H({\mathbf z}(t),t)+ F_{1t}({\mathbf q}(t),{\mathbf Q}(t),t)\ .$

This suggests defining the canonical transformation by the equations

(9)

$p=F_{1q}(q,Q,t)\ ,$

(10)

$P=-F_{1Q}(q,Q,t)\ .$

Owing to condition (5) and the inverse function theorem, (9) can be solved for $Q=\Phi_1(z,t)$ (at least locally in $\Omega$ ). Substitution of the solution in (7) gives $P=\Phi_2(z,t)$ as well. To get the inverse transformation $z=\Psi(Z,t)$ , solve (10) for $q=\Psi_1(Z,t)$ , then substitute in (9) to find $p=\Psi_2(Z,t)$ . Then the new Hamiltonian is defined by

(11)

$K(Z,t)=H(z,t)+F_{1t}(q,Q,t)=H(\Psi(Z,t),t)+F_{1t}(\Psi_1(Z,t),Q,t)\ .$

Textbooks usually apply a variational principle to show that the equations of motion are invariant in form under the transformation just defined. The advantage of the variational argument lies in its geometrical foundation, which motivates the starting equation (4), but is too long a story for this brief account; see Arnol'd (1974) for the geometric viewpoint. By generalizing an idea in Jacobi's 20th lecture (Jacobi (1842-1843), pp.158-159), the proof may be carried out instead by direct calculation. Substitution of (9) and (10) in (11) gives

(12)

$H(q,F_{1q}(q,Q,t),t)+F_{1t}(q,Q,t)=K(Q,-F_{1Q}(q,Q,t),t)\ .$

Take $\partial/\partial Q$ of (12), evaluate along orbits, and then subtract $d/dt$ of (7). Similarly, take $\partial/\partial q$ of (12), evaluate on orbits, and add $d/dt$ of (6). This leads to the informative equations

(13)

$F_{1qQ}(\dot{\mathbf q}-H_p)+(\dot{\mathbf P}+K_Q)+F_{1QQ}(\dot{\mathbf Q}-K_P)=0\ ,$

(14)

$F_{1qQ}(\dot{\mathbf Q}-K_P)-(\dot{\mathbf p}+H_q)+F_{1qq}(\dot{\mathbf q}-H_p)=0\ .$

In view of (5), this shows that (1) implies (2) and vice versa, as long as $(q,Q,t)$ lies in $\Omega$ .

There are other possible choices of the old and new variables on which the generator may depend. In general the condition (5) on $F_1(q,Q,t)$ will not hold globally, in which case one might try to use a function $F_2(q,P,t)$ with $\det F_{2qP}\ne 0$ . Then the equations analogous to (4), (9), 10), and (11) are

(15)

$p\dot q-H=-Q\dot P-K+dF_2/dt\ , \qquad p=F_{2q}\ ,\qquad Q=F_{2P}\ ,\qquad H+F_{2t}=K\ .$

One can show that the transformation induced by any generator with requisite smoothness is symplectic, which means that its Jacobian matrix $M=\{ \partial \Phi_i(z,t)/\partial z_j \}$ is symplectic for all $z$ . Written in terms of $n\times n$ blocks this condition is

(16)

$MJM^T=J\ ,\quad M=\begin{bmatrix}\partial Q/\partial q&\partial Q/\partial p\\ \partial P/\partial q&\partial P/\partial p\end{bmatrix}\ ,\quad J=\begin{bmatrix}0& -I\\I& 0\end{bmatrix}\ ,$

where $T$ denotes transpose. For $n=1$ the symplectic condition reduces to $\det M=1$ . To prove (16), differentiate (9) and (10) with respect to $q$ and $p$ . Thanks to (5), the resulting equations can be solved for $M$ ; some calculation then shows that the solution obeys (16). An alternative viewpoint is to take symplecticity as the defining property of a canonical transformation (Meyer & Hall,1991).

[edit]

Hamilton-Jacobi Equation and Invariant Tori

To produce a useful transformation the generator $F$ must be determined so that $K$ is indeed independent of $Q$ , thus giving (3) as the solution of the transformed equations. With this form of $K$ , substitution of (9) in (11) yields

(17)

$H(q, F_q(q,P,t),t)+ F_ t(q,P,t)=K(P,t)\ ,$

which is the Hamilton-Jacobi equation for the type-2 generator. Here $P$ is regarded as a (vector) parameter; the independent variables of the PDE are $q$ and $t$ . A solution of (17) depending on $n$ parameters $P_i$ and such that $\det F_{qP}\ne 0$ was called a complete solution (Vollständige Lösung) by Jacobi (in his case $K=0$ , and the $P_i$ are parameters with no particular interpretation). As was shown above, it determines a canonical transformation.

Equation (17) is clearly a necessary condition on the generator of a transformation for which the new Hamiltonian is independent of $Q$ . But can it be used to construct such a generator? For this a first question is the status of the function $K(P,t)$ . Can it be chosen arbitrarily or is it somehow determined in the course of the solution of (17)? There is one choice of $K$ which can be made freely at the start, and which leads to an important solution, namely $K= 0$ . This is the case considered by Hamilton and Jacobi, to be discussed in the following section.

To illustrate the situation with non-zero $K$ , take the case of a time-independent Hamiltonian $H(z)$ and look for a solution in which $K$ and $F$ are also time-independent. Take polar coordinates $(q,p)=(\phi,I), \ \ (Q,P)=(\psi,J)$ where $\phi,\psi\in [0,2\pi],\ \ I,J\in [0,\infty)$ . Also, define $G$ so that $F(\phi,J)=\phi\cdot J+G(\phi,J)$ , where the first term on the right gives the identity transformation. Then the Hamilton-Jacobi equation to solve for $G$ is

(18)

$H(\phi,J+G_\phi(\phi,J))=K(J)\ ,$

and the equations (9) and (10) defining the transformation are

(19)

$I=J+G_\phi(\phi,J)\ ,$

(20)

$\psi=\phi+G_J(\phi,J)\ .$

If $G$ satisfies (18) for some function $K(J)$ , then $J$ is constant and (19) represents an invariant torus in phase space. The new angle variable $\psi$ advances linearly in time, according to (3).

Now consider a perturbed integrable system with Hamiltonian

(21)

$H(\phi,I)=H_0(I)+\epsilon V(\phi,I)\ ,$

which satisfies a condition of non-degeneracy

(22)

$\det\ \nu_I(I)\ne 0,\quad \nu(I)= H_{0I}(I) .$

Next rearrange (18) to subtract the first terms of the Taylor series of $H_0(J+G_\phi)$ :

(23)

$-\nu(J)\cdot G_\phi=\epsilon V(\phi,J+G_\phi)+\big[ H_0(J+G_\phi)-H_0(J)-\nu(J)\cdot G_\phi\big] +\big[ H_0(J)-K(J)\big] \ .$

The sum of the terms in the first square bracket is $\mathcal{O}(G_\phi^2)$ and therefore small if the transformation (19,20) is close to the identity. Introduce the (multiple) Fourier series

(24)

$G(\phi,J)= \sum_{m\in Z^n} g_m(J)\exp(im\cdot\phi)\$

so that

(25)

$G_\phi(\phi,J)= \sum_{m\in Z^n} im\ g_m(J)\exp(im\cdot\phi)\ ,$

and take the Fourier transform of (23) to obtain

(26)

$g_m(J)=\frac{i}{m\cdot\nu(J)}\frac{1}{(2\pi)^n}\int_{T^n} \exp(-im\cdot\phi)\big[\epsilon V(\phi,J+G_\phi) + H_0(J+G_\phi)-H_0(J)-\nu(J)\cdot G_\phi\big]d\phi,\quad m\ne {\mathbf 0}\ .$

Since $G_\phi$ does not contain the zero mode, the set of equations (25) and (26) is a closed system for the Fourier coefficients $g_m,\ m\ne{\mathbf 0}$ . If a solution of this system is known for some $J$ , then the projection of (18) onto every mode except the zero mode has been solved. The zero mode projection is solved as well simply by defining $K$ as the average of the left-hand side:

(27)

$K(J)=\frac{1}{(2\pi)^n}\int_{T^n}d\phi\big[H_0(J+G_\phi)+ \epsilon V(\phi,J+G_\phi)\big]\ .$

This gives some understanding of how the PDE (18) could be solved without prior knowledge of its right-hand side. The zero mode amplitude $g_{\mathbf 0}$ can be chosen arbitrarily, for instance put equal to zero.

At first sight Eq.(26) would seem to be a straightforward fixed point problem that might be solved by some kind of iteration, provided that the divisor $m\cdot\nu(J)$ could be bounded away from zero through an appropriate choice of $J$ . Thanks to (22) the value of $\nu$ can be controlled by varying $J$ . The iteration might be started by keeping only the term $\epsilon V$ , which gives lowest order perturbation theory. If the series (24) is truncated, then the problem can indeed be approached in that way, and (26) provides a practical method for computing approximate invariant tori (Warnock & Ruth (1987)). The exact problem requires the refined method of KAM theory to control small divisors $m\cdot\nu$ for large $m$ (Gallavotti (1983), Pöschel (1982)). The theory ensures the existence of invariant tori for sufficiently small $\epsilon$ , but they are not continuous functions of $J$ . Rather, they exist only on a Cantor set in $J$ -space, and the concept of complete solution does not apply in the classical sense (it is nevertheless possible to construct a smooth function which solves the Hamilton-Jacobi equation on the above-mentioned Cantor set; see Pöschel (1982)).

[edit]

Action as a Solution of the Hamilton-Jacobi Equation

The following discussion is mostly an interpretation of Jacobi's 19th lecture. For a geometric approach see Arnol'd (1974), Section 46C. The goal is to solve the Hamilton-Jacobi equation for a Type-1 generator with the new Hamiltonian $K = 0$ . Write $Q=q_0$ so that the equation is

(28)

$H(q,F_{1q}(q,q_0,t),t)+F_{1t}(q,q_0,t)=0\ .$

Using the method of characteristics, suppose that the characteristic (orbit) ${\mathbf z}(t,z_0)=({\mathbf q}(t,z_0),{\mathbf p}(t,z_0))$ which solves (1) is known. Let us try to determine $F_1(q,q_0,t)$ from its values for $q={\mathbf q}(t,z_0)$ by means of an ODE for $g(t)=F_1({\mathbf q}(t,z_0),q_0,t)$ . Since $\dot g=F_{1q}\dot q+F_{1t}$ , equations () and (28) suggest putting

(29)

$\dot g(t)={\mathbf p}(t,z_0)\cdot \dot{\mathbf q}(t,z_0)-H({\mathbf z}(t,z_0),t)\ ,$

whence by integration the proposal

(30)

$F_1({\mathbf q}(t,z_0),q_0,t) = \int_0^t\big[{\mathbf p}(\tau,z_0)\cdot \dot {\mathbf q}(\tau,z_0)-H({\mathbf z} (\tau,z_0),\tau)\big]d\tau\ \equiv S(q_0,p_0,t)\ .$

From this one would like to get $F_1(q,q_0,t)$ for general $q$ , but that can be done only if $p_0$ can be deduced from the $2n+1$ numbers $(q,q_0,t)$ . In general this is not possible for all $t$ ; since orbits projected onto $q$ space can cross, there can be more than one $z_0$ giving the same ${\mathbf q}(t,z_0)$ . The locus of such crossings is called a caustic. To rule out caustics, the equation $q={\mathbf q}(t,q_0,p_0)$ must be solvable uniquely for $p_0=\mathcal{P}_0(q,q_0,t)$ . For that suppose $t>0$ and

(31)

$\det\bigg[\frac{\partial{\mathbf q }(t,q_0,p_0)}{\partial p_0}\bigg]\ne 0\ .$

Under these conditions the proposed generator is defined through (30) as

(32)

$F_1(q,q_0,t)=S(q_0,\mathcal{P}_0(q,q_0,t),t)\ .$

This was Hamilton's essential idea, to view the action (integral of the Lagrangian) as a function of initial and final coordinates and times.

To show that $F_1$ satisfies (28), first make a variation of the orbit, ${\mathbf z}(t,z_0)\rightarrow \tilde{\mathbf z}(t,\epsilon)={\mathbf z}(t,z_0)+\epsilon\delta{\mathbf z}(t)$ , where $\delta{\mathbf z}$ is an arbitrary $C^1$ function. After integration by parts the corresponding variation of (30) is

(33)

$\delta F_1({\mathbf q}(t,z_0),q_0,t) \equiv \bigg[\frac{d}{d\epsilon}\int_0^t\big[\tilde{\mathbf p}(\tau,\epsilon) \cdot\frac{d}{d\tau}\tilde{\mathbf q}(\tau,\epsilon)-H(\tilde{\mathbf z}(\tau,\epsilon),\tau)\big]d\tau \bigg]_{\epsilon=0} =\int_0^t\big[(\dot{\mathbf q}-H_p)\cdot\delta {\mathbf p}-(\dot{\mathbf p}+H_q)\cdot\delta{\mathbf q} \big]d\tau+{\mathbf p}(\tau,z_0)\cdot\delta{\mathbf q}(\tau)\bigg|_0^t\ .$

Since the integral is zero by (1), it follows that

(34)

$\delta F_1({\mathbf q}(t,z_0),q_0,t)=F_{1q}({\mathbf q}(t,z_0),q_0,t)\cdot\delta{\mathbf q}(t,z_0)+F_{1q_0}({\mathbf q}(t,z_0),q_0,t)\cdot\delta q_0= {\mathbf p}(t,z_0)\cdot\delta{\mathbf q}(t,z_0)-p_0\cdot\delta q_0\ ,$

and since the variations are arbitrary

(35)

${\mathbf p}(t,z_0)=F_{1q}({\mathbf q}(t,z_0),q_0,t)\ ,$

(36)

$p_0=-F_{1q_0}({\mathbf q}(t,z_0),q_0,t)\ .$

Next take $d/dt$ of (30) and apply (35) to obtain

(37)

$H({\mathbf q}(t,z_0),F_{1q}({\mathbf q}(t,z_0),q_0,t),t)+F_{1t}({\mathbf q}(t,z_0),q_0,t)=0\ .$

Now this shows that $F_1$ satisfies the Hamilton-Jacobi equation (28) since for any $q, q_0$ there is a $p_0$ such that $q={\mathbf q}(t,z_0)$ , by condition (31).

Recalling the equations () that define the canonical transformation, it is seen from (35) and (36) that the transformation from new to old variables is just the time evolution $z={\mathbf z}(t,z_0)$ , with the new variables being just the initial conditions $z_0$ , which are constant because the new Hamiltonian is zero. The condition $\det(F_{1qq_0})\ne 0$ is implied by (31) and (36), as may be seen by differentiating the latter with respect to $p_0$ , then taking determinants.

If it is not possible to solve $q={\mathbf q}(t,q_0,p_0)$ for $p_0$ , it may instead be possible to solve for $q_0=\mathcal{Q}_0(q,p_0,t)$ . Then we can use a generator of Type 2, easily constructed by a Legendre transformation of $F_1$ (Goldstein (1981)). Namely,

(38)

$F_2(q,p_0,t)=F_1(q,q_0,t)+q_0\cdot p_0 \equiv F_1(q,\mathcal{Q}_0(q,p_0,t),t)+ \mathcal{Q}_0(q,p_0,t)\cdot p_0=S(\mathcal{Q}_0(q,p_0,t),p_0,t)+\mathcal{Q}_0(q,p_0,t)\cdot p_0$ .

By again applying the variational argument, it is easy to check that $F_2$ satisfies all the required equations.

The discussion above proves existence of a solution of (28) in terms of the more elementary existence theory for (1), and also suggests methods of numerical solution of (28).

[edit]

Solution of classical problems by separation of variables

Hamilton's principal function (32) solves the Hamilton-Jacobi equation (28) and determines an infinite family of orbits, but in order to construct it one needs to know this family of orbits at the start. Gantmacher (1970, Chap.4, Sect.26) refers to this as a ``vicious circle", and states that ``Jacobi's contribution consists in the fact that he continued Hamilton's investigation and broke the vicious circle". He showed that any solution $F(q,t,\alpha)$ of

(39)

$H(q,F_q)+F_t=0\ ,$

depending on real parameters $\alpha= (\alpha_1,\cdots,\alpha_n)$ and complete in the sense that $\det F_{q\alpha}\ne 0$ , determines the orbits of the system.

It is instructive to illustrate Jacobi's program in a soluble example. Gantmacher (1970, Chap.4, Sect.27) describes three structures of the Hamiltonian, labeled $1^o, 2^o, 3^o$ for which (28) is explicitly soluble, each embodying the idea of ``separation of variables". Type $2^o$ includes some basic systems. For this case in two degrees of freedom the Hamiltonian has the form

(40)

$H(z)=H_2(H_1(z_1),z_2)\ ,\quad z_i=(q_i,p_i)\ ,$

and similarly for $n$ degrees of freedom,

(41)

$H(z)=H_n(\cdots H_3(H_2(H_1(z_1),z_2),z_3)\cdots,z_n)\ .$

Each $H_n$ is required to be $C^1$ in all arguments and to satisfy

(42)

$\frac{\partial H_n(\cdots,q_n,p_n)}{\partial p_n}\ne 0\ .$

Considering now two degrees of freedom, notice that because of (42) there exist functions $G_1,G_2$ such that

(43)

$H_1(q_1,G_1(q_1,\alpha_1))=\alpha_1\ ,\quad H_2(\alpha_1,q_2,G_1(q_2,\alpha_2,\alpha_1))=\alpha_2\ ,$

for any constants $\alpha_1,\alpha_2$ . Identification of $G_i$ with $F_{q_i}$ gives a solution of (39) in the form

(44)

$F(q,t,\alpha)=\int_{q_{10}}^{q_1}G_1(q_1^\prime,\alpha_1)dq_1^\prime + \int_{q_{20}}^{q_2}G_2(q_2^\prime,\alpha_2,\alpha_1)dq_2^\prime-\alpha_2 t$

Indeed, after substitution of this function and application of (43) the l.h.s. of (39) reads $\alpha_2-\alpha_2$ . Moreover, this solution is complete, as is seen by differentiating the first equation of (43) with respect to $\alpha_1$ and the second with respect to $\alpha_2$ . Because of (42) that shows that $D_2G_1\ne 0$ and $D_3G_2\ne 0$ hence $\det F_{q\alpha} =D_1G_1D_3G_2\ne0$ , where $D_i$ means partial derivative with respect to the $i$ -th argument.

Jacobi's crucial point is that the solution $q(t),p(t)$ of the equations

(45)

$p=F_q(q,t,\alpha)\ ,$

(46)

$\beta=F_\alpha(q,t,\alpha)\ ,$

is exactly the solution of Hamilton's equations, if $\beta=(\beta_1,\beta_2)$ is a constant. To prove this take $\partial/\partial\alpha$ of (39) and subtract $d/dt$ of (45). Also take $\partial/\partial q$ of (39) and subtract $d/dt$ of (46). Thus

(47)

$F_{q\alpha}(H_p-\dot q)=0\ ,\quad H_q+\dot p+F_{qq}(H_p-\dot q)=0\ .$

Since $F$ is a complete solution, the first equation implies $\dot q=H_p$ , and the second then implies $\dot p=-H_q$ . It was a generalization of this argument that provided Eqs. (13) and (14). The values of $(\alpha,\beta)$ to impose an arbitrary initial condition $(q(0),p(0))$ are obtained by solving (45), (46) at $t=0$ , again using completeness.

An example is planar motion in a central potential $V(r)$ (Goldstein, 1981). In polar coordinates the Hamiltonian for a particle of mass $m$ is

(48)

$H=\frac{1}{2m}\bigg[p_r^2+\frac{p_\phi^2}{r^2}\bigg]+V(r)\ , \quad (q_1,p_1)=(\phi,p_\phi)\ ,\quad (q_2,p_2)=(r,p_r)\ .$

Since $H$ is independent of $\phi$ , the conjugate momentum $p_\phi$ is constant in time; it is the conserved angular momentum. To apply the above scheme put

(49)

$H_1(q_1,p_1)=p_1=\alpha_1\ ,\quad H_2(\alpha_1,q_2,p_2)=\frac{1}{2m} \bigg[p_2^2+\frac{\alpha_1^2}{q_2^2}\bigg]+V(q_2)=\alpha_2\ .$

Notice that $H_2$ satisfies (42) if and only if $p_2\ne0$ . Now the $G_i$ defined by (43) are

(50)

$G_1(q_1,\alpha_1)=\alpha_1\ ,$

(51)

$G_2(q_2,\alpha_2,\alpha_1)=\pm\Pi(q_2,\alpha_2,\alpha_1)\ ,\quad \Pi= \bigg[2m\big(\alpha_2-V(q_2)\big)-(\alpha_1/q_2)^2\bigg]^{1/2}\ge 0$

In physicist's notation $\alpha_1=L=$ angular momentum, $\alpha_2=E=$ energy, and the formula (44) reads

(52)

$F(\phi,r,L,E,t)=L(\phi-\phi_0)-Et\pm\int_{r_0}^r\Pi(\rho,E,L)d\rho\ ,\quad \Pi(\rho,E,L)=\big[2m(E-V(\rho))-(L/\rho)^2\big]^{1/2}\ .$

Here $r_0, r$ must be such that the argument of the square root is non-negative in the region of integration. The motion $\phi(t),\ r(t)$ is obtained by solving (46). To that end compute

(53)

$\beta_1=F_L=\phi-\phi_0\mp L\int_{r_0}^r\Pi(\rho,E,L)^{-1}d\rho/\rho^2\ ,$

(54)

$\beta_2=F_E=-t\pm 2m\int_{r_0}^r\Pi(\rho,E,L)^{-1}d\rho\ .$

For initial conditions $\phi_0,\ p_{\phi_0},\ r_0,\ p_{r_0}$ the parameters are $\beta_1=0,\ \beta_2=0,\ \alpha_1=p_{\phi_0}=L,\ \alpha_2= (p_{r0}^2+(\alpha_1/r_0)^2)/2m+V(r_0)=E$ . Now (54) gives $t(r)$ , which must be inverted to give $r(t)$ ; then (53) gives $\phi(t)$ . This is the standard solution derived less elegantly in elementary treatments without the Hamilton-Jacobi method. The choice of sign in front of the integrals depends on $t$ and initial conditions. Suppose that the potential is attractive and $E$ is such that there is oscillatory motion in the effective one-dimensional potential with $r_0\le r\le r_1$ (Goldstein, 1981). During the first half-period $T/2$ the integral (54) runs from $r_0$ to $r\le r_1$ , with the plus sign. During the second half-period the integral is defined as $T/2$ plus the integral from $r_1$ to $r\ge r_0$ with the minus sign, and so on. Within any half-period the integral is monotonic in $r$ so that the inversion of $t(r)$ is always possible.

[edit]

Applications and Numerical Methods

There is a large literature on numerical solution of the Hamilton-Jacobi equation and similar equations, often applied to the special case of eikonal equations (Goldstein (1981), p.489), and ranging from classical approaches to generalized solutions of viscosity type (Lions (1982), Evans (2008)). For invariant tori in mechanics an extremely efficient method is to fit a Fourier series to orbits with non-resonant frequencies (Warnock & Ruth (1992)). Hamilton's Principal Function offers a way to construct the generator of symplectic time evolution maps for long time intervals, a valuable tool for modeling particle accelerators and other systems. For large $T$ one can evaluate $F_1(q,q_0,T)$ for $(q,q_0)$ on a finite mesh $\{ q_i,q_{0j} \}$ and then interpolate by B-splines of at least cubic degree, thereby getting a $C^2$ generator that will induce an exactly symplectic map. This approach, encouraged by the success of earlier similar work (Warnock & Berg (1997)), is under study.

[edit]

Bibliography

Arnol'd, V. I., "Mathematical Methods of Classical Mechanics", (Springer, New York, 1974).

Benton, S. H., "The Hamilton-Jacobi Equation: A Global Appraoch", (Academic Press, New York, 1977).

Born, M. and Wolf, E., "Principles of Optics", (Pergamon Press, Oxford, 1965)

Butterfield, J., "On Hamilton-Jacobi Theory as a Classical Root of Quantum Theory", in ``Quo Vadis Quantum Mechanics?", A. Elitzov et al., Eds. (Springer, New York, 2005).

Carathéodory, C., "Geometrische Optik", (Springer, Berlin, 1937).

Carathéodory, C., "Calculus of Variations and Partial Differential Equations of the First Order", (Chelsea, New York, 1982).

Chapman, S., Garrett, B. C., and Miller, W. H., "Semiclassical Eigenvalues for Nonseparable Systems: Nonperturbative Solution of the Hamilton-Jacobi Equation in Action-Angle Variables", J. Chem. Phys. 64, 502-509 (1976).

Courant, R. and Hilbert, D., "Methods of Mathematical Physics, Vol. II", (Interscience, New York, 1962.

Evans, L., "Weak KAM theory and partial differential equations", in Calculus of Variations and Nonlinear Partial Differential Equations, pp. 123-154, Lecture Notes in Math. 1927 (Springer, Berlin, 2008).

Fleming, W. H. and Rishel, R. "Deterministic and Stochastic Optimal Control", (Springer, Berlin, 1975)

Gallavotti, G., "The Elements of Mechanics", (Springer, New York, 1983).

Gantmacher, F., "Lectures in Analytical Mechanics", (MIR Publishers, Moscow, 1970).

Goldstein, H., "Classical Mechanics", (Addison-Wesley, Menlo Park, 1981).

Hamilton, W. R., "The Mathematical Papers of William Rowan Hamilton, Vol. I, Geometrical Optics, Vol.II, Dynamics" (Cambridge University Press, 1931), especially three Supplements (1830-1832) to the "Theory of Systems of Rays" (1827) in Vol.I, and "On a General Method in Dynamics" (1832) in Vol.II.

Jacobi, C. G. J., "Vorlesungen über Dynamik", Königsberg lectures of 1842-1843, (reprinted by Chelsea Publishing Co., New York, 1969).

Lanczos, C., "The Variational Principles of Mechanics", (U. Toronto Press, Toronto, 1949).

Landau, L. D. and Lifshitz, E. M., "Mechanics", (Pergamon Press, Oxford, 1969).

Lions, P.-L., "Generalized Solutions of Hamilton-Jacobi Equations", (Pitman, Boston, 1982)

Martens, C. C. and Ezra, G. S., "Semi-classical Mechanics of Strongly Resonant Systems: a Fourier Transform Approach", J. Chem. Phys. 86, 279-307 (1987).

K. Meyer and G. Hall, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem", (Springer, New York, 1991).

Nekhoroshev, N. N., "An Exponential Estimate of the Time of Stability of Nearly Integrable Hamiltonian Systems", Russ. Math. Surveys 32, 6, 1-65 (1977).

Percival, I. C., "Semiclassical Theory of Bound States", in Advances in Chemical Physics 36 (Wiley, New York, 1977).

Pöschel, J., "Integrability of Hamiltonian Systems on Cantor Sets", Comm. Pure Appl. Math. 35, 653-695 (1982).

Synge, J. L., "Geometrical Optics, an Introduction to Hamilton's Method", (Cambridge University Press, 1937).

Warnock, R. and Ruth, R. D., "Long Term Bounds on Nonlinear Hamiltonian Motion", Physica D 56 188-215 (1992).

Warnock, R. and Ruth, R. D., "Invariant Tori through Direct Solution of the Hamilton-Jacobi Equation", Physica D 26, 1-36 (1987).

Warnock, R. and Berg, J. S., "Fast Symplectic Mapping and Long-term Stability Near Broad Resonances", AIP Conf. Proc. 395 (Amer. Inst. Phys., 1997).

==See also==</ref>

Suggested by:	Prof. James Meiss, Applied Mathematics University of Colorado
Invited by:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Action editor:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Reviewer B:	Dr. H. Scott Dumas, Dept. of Mathematical Sciences, University of Cincinnati

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Categories: Dynamical Systems

Hamilton-Jacobi equation

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Canonical Transformation

Hamilton-Jacobi Equation and Invariant Tori

Action as a Solution of the Hamilton-Jacobi Equation

Solution of classical problems by separation of variables

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