Robert L. Warnock
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==Bibliography== | ==Bibliography== | ||
| − | Arnol'd, V. I., | + | Arnol'd, V. I., "Mathematical Methods of Classical Mechanics", (Springer, New York, 1974). |
| − | Benton, S. H., | + | Benton, S. H., "The Hamilton-Jacobi Equation: A Global Appraoch", (Academic Press, New York, 1977). |
| − | Born, M. and Wolf, E., | + | Born, M. and Wolf, E., "Principles of Optics", (Pergamon Press, Oxford, 1965) |
| − | Butterfield, J., | + | Butterfield, J., "On Hamilton-Jacobi Theory as a Classical Root of Quantum Theory", |
in ``Quo Vadis Quantum Mechanmics?", A. Elitzov et al., Eds. (Springer, New York, 2005). | in ``Quo Vadis Quantum Mechanmics?", A. Elitzov et al., Eds. (Springer, New York, 2005). | ||
| − | Carathe'odory, C., | + | Carathe'odory, C., "Geometrische Optik", (Springer, Berlin, 1937). |
| − | Carathe'odory, C., | + | Carathe'odory, C., "Calculus of Variations and Partial Differential Equations |
of the First Order", (Chelsea, New York, 1982). | of the First Order", (Chelsea, New York, 1982). | ||
| − | Courant, R. and Hilbert, D., | + | Courant, R. and Hilbert, D., "Methods of Mathematical Physics, Vol. II", |
(Interscience, New York, 1962. | (Interscience, New York, 1962. | ||
| − | Fleming, W. H. and Rishel, R. | + | Fleming, W. H. and Rishel, R. "Deterministic and Stochastic Optimal Control", |
(Springer, Berlin, 1975) | (Springer, Berlin, 1975) | ||
| − | Gallavotti, G., | + | Gallavotti, G., "The Elements of Mechanics", (Springer, New York, 1983). |
| − | Gantmacher, F., | + | Gantmacher, F., "Lectures in Analytical Mechanics", (MIR Publishers, Moscow, 1970). |
| − | Goldstein, H., | + | Goldstein, H., "Classical Mechanics", (Addison-Wesley, Menlo Park, 1981). |
| − | Hamilton, W. R., | + | 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 | + | (Cambridge University Press, 1931), especially three Supplements (1830-1832) to the "Theory of Systems of Rays" |
| − | (1827) in Vol.I, and | + | (1827) in Vol.I, and "On a General Method in Dynamics" (1832) in Vol.II. |
| − | Jacobi, C. G. J., | + | Jacobi, C. G. J., "Vorlesungen ueber Dynamik", Koenigsberg lectures of 1842-1843, |
(reprinted by Chelsea Publishing Co., New York, 1969). | (reprinted by Chelsea Publishing Co., New York, 1969). | ||
| − | Lanczos, C., | + | Lanczos, C., "The Variational Principles of Mechanics", (U. Toronto Press, Toronto, 1949). |
| − | Landau, L. D. and Lifshitz, E. M., | + | Landau, L. D. and Lifshitz, E. M., "Mechanics", (Pergamon Press, Oxford, 1969). |
| − | Lion, P. L., | + | Lion, P. L., "Generalized Solutions of Hamilton-Jacobi Equations", (Pitman, Boston, 1982) |
| − | Nekhoroshev, N. N., | + | Nekhoroshev, N. N., "An Exponential Estimate of the Time of Stability of |
Nearly Integrable Hamiltonian Systems", Russ. Math. Surveys '''32''', 6, 1-65 (1977). | Nearly Integrable Hamiltonian Systems", Russ. Math. Surveys '''32''', 6, 1-65 (1977). | ||
| − | Percival, I. C., | + | Percival, I. C., "Semiclassical Theory of Bound States", in Advances in Chemical Physics '''36''' (Wiley, New York, 1977). |
| − | Poeschel, J., | + | Poeschel, J., "Integrability of Hamiltonian Systems on Cantor Sets", Comm. Pure Appl. Math. '''35''', 653-695 (1982). |
| − | Synge, J. L., | + | Synge, J. L., "Geometrical Optics, an Introduction to Hamilton's Method", (Cambridge University Press, 1937). |
| − | Warnock, R. and Ruth, R. D., | + | 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., | + | Warnock, R. and Ruth, R. D., "Invariant Tori through Direct Solution of |
the Hamilton-Jacobi Equation", Physica D '''26''', 1-36 (1987). | the Hamilton-Jacobi Equation", Physica D '''26''', 1-36 (1987). | ||
| − | Warnock, R. and Berg, J. S., | + | Warnock, R. and Berg, J. S., "Fast Symplectic Mapping and |
Long-term Stability Near Broad Resonances", AIP Conf. Proc. '''395''' (Amer. Inst. Phys., 1997). | Long-term Stability Near Broad Resonances", AIP Conf. Proc. '''395''' (Amer. Inst. Phys., 1997). | ||
Revision as of 21:21, 13 May 2009
The Hamilton-Jacobi Equation
Contents |
History
<label>history</label>Hamilton made one of the earliest studies of geometrical optics in an arbitrary medium with varying index of refraction (Hamilton, (1830-1832), 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).
With an emphasis on mechanics, Jacobi (1842-1843) 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 Lion (1982) and 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; 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)). Various forms and generalizations of the Hamilton-Jacobi equation occur widely in contemporary applied mathematics, for instance in optimal control theory (Fleming & Rishel (1975)).
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 \(t_0=0\).
One seeks a transformation of coordinates, \(Z=(Q,P)=\Phi(z,t)=(\Phi_1(z,t),\Phi_2(z,t))\), 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 \(z=\Psi(Z,t)\).
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
- <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. By writing out \(dF/dt\), it is seen 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 suggests the definition of 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, (<ref>canona</ref>) can be solved for \(P=\Phi_2(z,t)\) (at least locally in \(\Omega\)). Substitution of the solution in (<ref>canon2</ref>) gives \(Q=\Phi_1(z,t)\) as well. To get the inverse transformation \(z=\Psi(Z,t)\) 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>
Textbooks usually apply a variational principle to show that the equations of motion are invariant in form under the transformation just defined. By generalizing an idea in Jacobi's 20th lecture (Jacobi, 1842-1843), pp.158-159, it is possible to make the proof by a direct calculation. Substitution of (<ref>canona</ref>) and (<ref>canonb</ref>) in (<ref>canonc</ref>) gives
- <math explicit>
H(q,F_q(q,P,t),t)+F_t(q,P,t)=K(F_P(q,P,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) labels four choices, \(F_1(q,Q,t),\ F_2(q,P,t),\ F_3(p,Q,t),\ F_4(p,P,t)\); there are others in which different components of coordinates and momenta are treated differently. In each case we have \(K=H+\partial F_i/\partial t\), but the equations analogous to (<ref>canona</ref>) and (<ref>canonb</ref>) vary in form. The list of transformations is as follows:
- <math F1>
p\dot q-H=P\dot Q-K+dF_1/dt, \quad p= F_{1q},\quad P=-F_{1Q}\ ,
</math>
- <math F2>
p\dot q-H=-Q\dot P-K+dF_2/dt, \quad p=F_{2q},\quad Q=F_{2P}\ ,
</math>
- <math F3>
-q\dot p-H=P\dot Q-K+dF_3/dt, \quad q=-F_{3p},\quad P=-F_{3Q}\ ,
</math>
- <math F4>
-q\dot p-H=-Q\dot P-K+dF_4/dt, \quad q=-F_{4p},\quad Q=F_{4P}\ .
</math> In each case the determinant of crossed second derivatives of \(F_i\) should be non-zero, as in (<ref>nonsing</ref>). In addition to \(F=F_2\) treated above, we shall be interested in \(F_1\).
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>
To make a useful transformation the generator \(F\) must be determined in such a way that \(K\) is indeed independent of \(Q\), thus giving (<ref>soln</ref>) as the solution of the transformed equations. With that form of \(K\), substitution of (<ref>canona</ref>) in (<ref>canonc</ref>) yields
- <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 \(K=0\), and the \(P_i\) are parameters with no particular interpretation). As was shown above, 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, 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 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. Introduce the Fourier series
- <math fourier>
G_\phi(\phi,J)= \sum_{m\in Z^n} im\ g_m(J)\exp(im\cdot\phi)\ ,
</math>
and take the Fourier transform of (<ref>pertform</ref>) to obtain
- <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>) and (<ref>fourier</ref>) 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 (<ref>hjpolar</ref>) 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 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>
This gives some understanding of how the PDE (<ref>hjpolar</ref>) could be solved without a 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.(<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 \(m\cdot\nu \) at large \(m\). (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. 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 zero for the new Hamiltonian. 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>
Following 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 (<ref>hameq</ref>) 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 (<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>
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. To rule out caustics, it is required that the equation \(q={\mathbf q}(t,q_0,p_0)\) be solvable uniquely for \(p_0=\mathcal{P}_0(q,q_0,t)\). For that suppose \(t>0\) and
- <math nocaustic>
\det\bigg[\frac{\partial{\mathbf q }(t,q_0,p_0)}{\partial p_0}\bigg]\ne 0\ .
</math>
Under these conditions the proposed generator is defined through (<ref>actionint</ref>) as
- <math F1def>
F_1(q,q_0,t)=S(q_0,\mathcal{P}_0(q,q_0,t),t)\ .
</math>
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 (<ref>hj0</ref>) 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 an integration by parts the corresponding variation of (<ref>actionint</ref>) is
- <math deltaF>
\delta F_1({\mathbf q}(t,z_0),q_0,t)=:\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\ .
</math>
Since the integral is zero by (<ref>hameq</ref>) it follows that
- <math deltas>
\delta F_1({\mathbf q}(t,z_0),q_0,t)={\mathbf p}(t,z_0)\cdot\delta{\mathbf q}(t,z_0)-p_0\cdot\delta q_0
= 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\ ,
</math>
and since the variations are arbitrary
- <math pFq>
{\mathbf p}(t,z_0)=F_{1q}({\mathbf q}(t,z_0),q_0,t)\ ,
</math>
- <math p0Fq0>
p_0=-F_{1q_0}({\mathbf q}(t,z_0),q_0,t)\ .
</math> Next take \(d/dt\) of (<ref>actionint</ref>) and apply (<ref>pFq</ref>) to obtain
- <math hj0orb>
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\ .
</math>
Now this shows that \(F_1\) satisfies the Hamilton-Jacobi equation (<ref>hj0</ref>) since for any \(q, q_0\) there is a \(p_0\) such that \(q={\mathbf q}(t,z_0)\), by condition (<ref>nocaustic</ref>).
Recalling the equations (<ref>F1</ref>) that define the canonical transformation, it is seen from (<ref>pFq</ref>) and (<ref>p0Fq0</ref>) 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 (<ref>nocaustic</ref>) and (<ref>p0Fq0</ref>), as is 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,
- <math legendre>F_2(q,p_0,t)=F_1(q,q_0,t)+q_0\cdot p_0=: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 </math>. By again applying the variational argument, it is easy to check that \(F_2\) satisfies all the required equations.
Numerical Methods
There is a big 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 (Lion (1982)). 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.
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