Robert L. Warnock

Revision as of 04:57, 28 April 2009

History

Hamilton made one of the earliest studies of geometrical optics in an arbitrary medium with varying index of refraction (Hamilton, 1827), (Synge, 1937), (Carathe'odory, 1937). He found an eloquent summary 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 (or particle orbits in the extension to mechanics).

Jacobi (Jacobi,1842) sharpened Hamilton's formulation, clarified mathematical issues, and made significant applications. 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), (Carathe'odory, 1982), (Courant-Hilbert, 1962). For studies using modern PDE theory see (Benton, 1977). The theory embodies a wave-particle duality, which figured in the advent of the de Broglie - Schroedinger wave mechanics (Butterfield, 2005).

In a broader view 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 not only determine 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 (Gallavotti, 1983}, which has implications for stability of motion. Even approximate invariants have implications for stability over finite times (Nekhoroshev, 1977), (Warnock-Ruth, 1992), and also find applications in semi-classical quantum theory (Einstein-Brillouin-Keller quantization) (Percival, 1977). Various forms and generalizations of the Hamilton-Jacobi equation occur widely in contemporary applied mathematics, for instance in optimal control theory (Bellman, 1957).

Canonical Transformation <label>canon_sect</label>

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 ordinary differential equations,

<math hameq>

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

</math> 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 (<ref>hameq</ref>) 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 \(t_0=0\).

We seek a change of variable, \(Z=(Q,P)=\Phi(z,t)=(\Phi_1(z,t),\Phi_2(z,t))\), in general time dependent, so that the equations of motion retain their form, but with a new Hamiltonian \(K\), namely

<math newhameq>

   \dot Q= K_P(Z,t)\ ,\quad \dot P=- K_Q(Z,t)\ .
 </math>

If it can be arranged that \(K\) is independent of \(Q\), then \(\mathbf P\) is constant and the solution of (<ref>newhameq</ref>) is given simply as

<math soln>

  {\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\ .

</math> The solution of (<ref>hameq</ref>) is retrieved by the inverse transformation \(z=\Psi(Z,t)\).

We 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 \(Z_0=\Phi(z_0,0)\). We often suppress reference to initial conditions. A canonical transformation will be determined through the equation

<math canon>

 {\mathbf p}(t)\cdot\dot{\mathbf q}(t)-H({\mathbf z}(t),t)=-{\mathbf Q}(t)\cdot\dot{\mathbf P}(t)-K({\mathbf Z}(t),t)

+\frac{d}{dt}F({\mathbf q}(t),{\mathbf P}(t),t)\ , </math> where \(\cdot\) indicates the scalar product and the given function \(F(q,P,t)\) is \(C^2\) in its first two arguments, \(C^1\) in \(t\), and such that

<math nonsing>

 \det F_{qP}=\det\{\partial^2 F/\partial q_i\partial P_j\}\ne 0\ ,
 </math>

in some open region \(\Omega\in \mathbb{R}^{2n+1}\) of \((q,P,t)\)-space. This function \(F\) is called the generator or generating function of the transformation. Writing out \(dF/dt\), we see that (<ref>canon</ref>) is satisfied if

<math canon1>

{\mathbf p}(t)=F_q({\mathbf q}(t),{\mathbf P}(t),t)\ ,

</math>

<math canon2>

  {\mathbf Q}(t)=F_P({\mathbf q}(t),{\mathbf P}(t),t)\ ,

</math>

<math canon3>

  K({\mathbf Z}(t),t)=H({\mathbf z}(t),t)+ F_t({\mathbf q}(t),{\mathbf P}(t),t)\ .

</math> This leads us to define the canonical transformation by the equations

<math canona>

  p=F_q(q,P,t)\ ,

</math>

<math canonb>

  Q=F_P(q,P,t)\ .

</math> Owing to condition (<ref>nonsing</ref>) and the inverse function theorem we can solve (<ref>canona</ref>) for \(P=\Phi_2(z,t)\) (at least locally in \(\Omega\)) and then substitute in (<ref>canon2</ref>) to get \(Q=\Phi_1(z,t)\) as well. Similarly, to get the inverse transformation \(z=\Psi(Z,t)\) we solve (<ref>canonb</ref>) for \(q=\Psi_1(Z,t)\) then substitute in (<ref>canona</ref>) to find \(p=\Psi_2(Z,t)\). Then the new Hamiltonian is defined by <math canonc>

  K(Z,t)=H(z,t)+F_t(q,P,t)=H(\Psi(Z,t),t)+F_t(\Psi_1(Z,t),P,t)\ .

</math>

Take \(\partial/\partial P\) of (<ref>explicit</ref>), evaluate on orbits, and then subtract \(d/dt\) of (<ref>canon2</ref>). Similarly, take \(\partial/\partial q\) of (<ref>explicit</ref>), evaluate on orbits, and subtract \(d/dt\) of (<ref>canon1</ref>). This leads to the informative equations

<math jac20_1>

   F_{qP}(\dot{\mathbf q}-H_p)-(\dot{\mathbf Q}-K_P)+F_{PP}(\dot{\mathbf P}+K_Q)=0\ ,

</math>

<math jac20_2>

   F_{qP}(\dot{\mathbf P}+K_Q)-(\dot{\mathbf p}+H_q)+F_{qq}(\dot{\mathbf q}-H_p)=0\ .

</math> In view of (<ref>nonsing</ref>), this shows that (<ref>hameq</ref>) implies (<ref>newhameq</ref>) and vice versa, as long as \((q,P,t)\) lies in \(\Omega\).

There are other possible choices of the old and new variables on which the generating function depends (Goldstein,1981). Following notation of Goldstein, we are concerned with \(F_1(q,Q,t)\) and \(F= F_2(q,P,t)\), for which the equations are as follows:

<math F1>

    {\mathbf p }\dot{\mathbf q}-H={\mathbf P}\dot{\mathbf Q}-K+dF_1/dt, \quad p= F_{1q},\quad P=-F_{1Q}\ ,

</math>

<math F2>

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

</math> In each case \(K=H+F_{it}\) and the determinant of second derivatives of \(F_i\) should be non-zero, as in (<ref>nonsing</ref>).

One can show that the transformation defined 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 \(z\). An alternative viewpoint is to take symplecticity as the defining property of a canonical transformation.

Hamilton-Jacobi Equation and Invariant Tori <label>hjsect</label>

We now wish to determine \(F\) in such a way that \(K\) will indeed be independent of \(Q\), thus giving (<ref>soln</ref>) as the solution of the transformed equations. Demanding that form of \(K\), and substituting (<ref>canona</ref>) in (<ref>canonc</ref>) we have <math hj>

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

</math> which is the Hamilton-Jacobi equation for the type-2 generator. Here \(P\) is regarded as a parameter; the independent variables of the PDE are \(q\) and \(t\). A solution of (<ref>hj</ref>) depending on \(n\) parameters \(P_i\) and such that \(\det F_{qP}\ne 0\) was called a complete solution (Vollstaendige Loesung) by Jacobi; in his case \(K=0\). As we have seen, it determines a canonical transformation.

Equation (<ref>hj</ref>) 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 (<ref>hj</ref>)? 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, which we shall discuss 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 \((q,p)=(\phi,I), \ \ (Q,P)=(\psi,J),\ \ \phi,\psi\in [0,2\pi],\ \ I,J\in [0,\infty)\). Also, define \(G\) so that \(F(\phi,J)=\phi\cdot J+G(\phi,J)\), where the first term on the right gives the identity transform. Then the Hamilton-Jacobi equation to solve for \(G\) is

<math hjpolar>

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

</math> and the equations (<ref>canona</ref>) and (<ref>canonb</ref>) defining the transformation are

<math polara>

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

</math>

<math polarb>

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

</math> If \(G\) satisfies (<ref>hj</ref>) for some function \(K(J)\), then \(J\) is constant and (<ref>polara</ref>) represents an invariant torus in phase space. The new angle variable \(\psi\) advances linearly in time, according to (<ref>soln</ref>).

Now consider a perturbed integrable system with Hamiltonian

<math pertint>

     H(\phi,I)=H_0(I)+\epsilon V(\phi,I)\ ,
     </math>

which satisfies a condition of non-degeneracy

<math nondegen>

     \det\ \nu_I(I)\ne 0,\quad \nu(I)= H_{0I}(I) .
     </math>

Next rearrange (<ref>hjpolar</ref>) to subtract the first terms of the Taylor series of \(H_0(J+G_\phi)\):

<math pertform>

     -\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] \ .
     </math>

The sum of the terms in the first square bracket is \(\mathcal{O}(G_\phi^2)\) and therefore small if the transformation (<ref>polara</ref>,<ref>polarb</ref>) is close to the identity. Introducing the Fourier series

<math fourier>

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

and taking the Fourier transform of (<ref>pertform</ref>) we have

<math hjfourier>

      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}\ .
     </math>

Since \(G_\phi\) does not contain the zero mode, the set of equations (<ref>hjfourier</ref>) for all \(m\ne {\mathbf 0}\) 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 we have solved the projection of (<ref>hjpolar</ref>) onto every mode except the zero mode. We can then solve also the zero mode projection simply by defining \(K\) as the average of the l.h.s.:

<math Kdef>

     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]\ .
     </math>

The zero mode amplitude \(g_{\mathbf 0}\) can be chosen arbitrarily, for instance put equal to zero. Thus we get some understanding of how the PDE (<ref>hjpolar</ref>) could be solved without a prior knowledge of its right hand side.

At first sight Eq.(<ref>hjfourier</ref>) 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 \(J\). The iteration might be started by keeping only the term \(\epsilon V\), which gives lowest order perturbation theory. If the series (<ref>fourier</ref>) is truncated, then the problem can indeed be approached in that way, and (<ref>hjfourier</ref>) 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 (Gallavotti, 1983), (Poeschel, 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 a literal sense (Poeschel, 1982)).

Action as a Solution of the Hamilton-Jacobi Equation <label>action_integral</label>

The following discussion is mostly an interpretation of Jacobi's 19th lecture (Jacobi, 1843). For a geometric approach see (Arnol'd, 1974), Section 46C. We wish to solve the Hamilton-Jacobi equation for a Type-1 generator with zero for the new Hamiltonian. We write \(Q=q_0\) so that the equation is

<math hj0>

H(q,F_{1q}(q,q_0,t),t)+F_{1t}(q,q_0,t)=0\ .
</math>

We proceed by the method of characteristics, supposing that the characteristic (orbit) \({\mathbf z}(t,z_0)=({\mathbf q}(t,z_0),{\mathbf p}(t,z_0))\) which solves (<ref>hameq</ref>) is known. We try to determine \(F_1(q,q_0,t)\) from its values for \(q={\mathbf q}(t,z_0)\), and therefore posit 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 (<ref>F1</ref>) and (<ref>hj0</ref>) suggest putting

<math dotg>

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

</math> whence by integration the proposal

<math actionint>

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\ =: S(q_0,p_0,t)\ .

</math>

We have chosen to take \(F_1(q_0,q_0,0)=0\). Since the r.h.s. depends only on \(z_0\) and \(t\), this makes sense only if it is possible to deduce \(p_0\) from the \(2n+1\) numbers \((q,q_0,t)\). In general that 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.