Invariants to All Orders in Classical Perturbation Theory

The method of averaging is applied to systems having Hamiltonians of the form $${\mathrm{H=H}}_0(\mathbf{J}{\mathrm{)+\epsilon H}}_1(\mathbf{J,\psi ,p,q})$$ , where H₁ is periodic in each of the components of Ψ. When the system is nondegenerate it is shown that corresponding to each component of Ψ there is a quantity K_i which is invariant to all orders in ε. When the system has an m‐fold degeneracy, somewhat weaker results are obtained. In this case it is shown that the Hamiltonian, when expressed in terms of the average variables, depends on the angle variables only through their m degenerate combinations. This is true to all orders in ε. Thus, if Ψ has s components there are s ‐ m invariants provided that the average variables can be made canonical. However, the conditions under which degenerate perturbation theory can be made canonical are not known. The invariants which arise when the Hamiltonian has an adiabatic or a harmonic time dependence are also discussed. The techniques are applied to the simple case of a harmonic oscillator whose frequency varies slowly with time.