Hamiltonian Equations Of Motion Research Articles

Wave Propagation and a Secular Theory of Calculation Based on Hamilton-Jacobi Theory

Summary A Hamilton-Jacobi method, often used in Astronomy, is outlined. It allows one to suppress secular terms in perturbation calculations. Noting the correspondence between classical mechanics and wave propagation, we use the method to treat the effect of small lateral variations in structure on the propagation of elastic surface waves. The basic result is that the horizontal wave vector is conserved, but the change in position, at which a particular spectral density is found, depends linearly on time. The purpose of this note is to outline a Hamilton-Jacobi method for the solution of wave propagation with perturbations acting (Osgood 1965; Smart 1953). This method has been used in astronomical calculations but has not yet received notice from the geophysical community. The basic result is the following. For a physical system with Hamiltonian Ho-H,, it is possible to obtain a set of variables (a, B) which satisfy Hamiltonian equations in which the new Hamiltonian is HI. When HI is zero, these variables become constants of the motion for the system with Hamiltonian H,. Several examples using this method will be shown (Bogoliubov & Mitropolsky 1961; Backus 1962a, b). For a physical system with Hamiltonian H,, the Hamiltonian equations of motion are with q and p the co-ordinates and momenta respectively

Separation of Motions of Many-Body Systems into Dynamically Independent Parts by Projection onto Equilibrium Varieties in Phase Space. II

A condensed notation in phase space that facilitates calculations is developed. With the aid of this notation, the results are given a simple geometrical interpretation in phase space by introducing a certain canonically invariant metrical tensor O/sub ij/. This tensor does ot yield the usual orthogonal or pseudo-orthogonal metric, but rather, what is called a symplectic metric. One then sees that the projections that are made are orthogonal, in the symplectic sense, to the equilibrium varieties. Likewise, one can see quite generally, that the entire canonical formalism, including the Poisson brackets and the hamiltonian equations of motion, reduces to simple geometrical relations in phase space, the form of which is suggestive for possible (urther developments, especially with regard to the treatment in higher approximations. The ideas are applied to the electron gas, and the dynamics of the plasma are illustrated with the aid of a comparison with a simple two-dimensional model, possessing all the essential features described above. In this way, one can understand many of the basic features of the plasma motions, in terms of concepts such as the generalization of the notion of centrifugal force and Coriolis force to phase space. By going over to a local geodeticmore » frame in the equilibrium variety, one is led in a natural way to the concept of a set of quasiparticles for the plasma. If the number of collective oscillatory coordinates is s, then there will be 3N-s of these quasiparticle coordinates. The latter do not represent any of the actual original particles out of which the system is constituted, but rather, they represent effective pulse-like distributions of charge, which move together in a correlated way so as to resemble an actual particle in many respects. (auth)« less

A note on the Hamiltonian equations of motion

The Eulerian equations obtained by varying a Lagrangian containing certain field quantities qα and their derivatives of any order are cast into canonical forms. A modified canonical form for the equation of motion is given when the conjugate variables are not independent. Finally a special case is discussed.