Robert L. Warnock
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Various forms and generalizations of the H-J equation occur widely in contemporary applied mathematics, | Various forms and generalizations of the H-J equation occur widely in contemporary applied mathematics, | ||
for instance in optimal control theory (Bellman 1957). | for instance in optimal control theory (Bellman 1957). | ||
| + | |||
| + | ==Canonical Transformation <label>canon_sect</label>== | ||
| + | |||
| + | A mechanical system with <math>n</math> degrees of freedom is described by generalized coordinates <math>q=(q_1,\cdots, q_n)</math> and corresponding | ||
| + | generalized momenta <math> p=(p_1,\cdots,p_n)</math>; we write <math>z=(q,p)</math>. 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 <math> \dot{}\ </math> denotes the time derivative and subscripts indicate vectors of partial derivatives; thus <math>H_q=(\partial H/\partial q_1,\cdots,\partial H/\partial q_n)</math>. | ||
| + | The Hamiltonian function <math>H:\mathbb{R}^{2n}\times\mathbb{R}\rightarrow\mathbb{R}</math> | ||
| + | is here assumed to be <math>C^2</math> in <math>z</math> and continuous in <math>t</math>. The solution of the initial value problem for the Hamiltonian system | ||
| + | (<ref>hameq</ref>) is denoted by <math>{\mathbf z}(t,z_0)=({\mathbf q}(t,z_0),{\mathbf p}(t,z_0))</math> for initial value <math>z_0={\mathbf z}(0,z_0)</math>. | ||
| + | This solution, denoted by the bold faced letter <math>\mathbf z</math> to distinguish it from | ||
| + | a general point <math>z</math> in phase space, will be called an ``orbit". If <math>H</math> depends on the time, specification | ||
| + | of an orbit requires the initial time <math>t_0</math> (not just the elapsed time) as well as the initial condition <math>z_0</math>; for convenience we choose the origin | ||
| + | of time so that <math>t_0=0</math>. | ||
Revision as of 21:27, 17 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 Hamilton and Jacobi, a solution of their PDE 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 H-J 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 we choose the origin
of time so that \(t_0=0\).
