Asymptotic Theory of Hamiltonian and other Systems with all Solutions Nearly Periodic

Abstract

Consider a system of N ordinary first-order differential equations in N dependent variables, and let the independent variable s not appear explicitly. Let the system depend on a small parameter ε and possess a formal infinite power series expansion in ε, and suppose that the limiting system for ε = 0 exists and has only periodic solutions. Then a formal solution can be constructed involving infinite power series in ε and satisfying the equations over large domains of s (of order 1/ε). The true solutions of the system exist over such domains and are asymptotically represented as ε→0 by the formal solutions. The construction is based on the standard type of formal series solution of a ``reduced'' system of N − 1 equations in N − 1 dependent variables and with the new independent variable σ = ε s; the omitted variable is essentially an angle variable φ describing the phase around the simple, closed curves. If the original system is Hamiltonian, then one can define the usual action integral J = ∫ p·dq to all orders; the integral is taken around the phase ring. It is proved that J is an integral of the system and that the Poisson bracket of φ with J is unity, both to all orders. The usefulness of this particular integral is that it is computable locally. The reduced system, after elimination of another dependent variable by means of the constancy of J, can itself be put in Hamiltonian form; if its solutions are nearly periodic, the whole procedure can be reapplied. The present theory encompasses previous proofs of adiabatic invariance to all orders for particular systems such as the harmonic oscillator, the nonlinear oscillator, the charged particle gyrating tightly in a given electromagnetic field, and the longitudinal back-and-forth motion of such a particle trapped between two ``magnetic mirrors'' in a weak electric field. There are many other applications.