Approximate invariants and Lagrangians for autonomous, weakly non-linear systems .2. Linear friction

In a previous paper, a procedure was given for the calculation of approximate, time-independent invariants for autonomous, weakly-non-linear systems with one degree of freedom, based on an expansion in linear and non-linear parameters. Such an invariant generates an approximate, time-independent Lagrangian and Hamiltonian describing the system. An extension of this procedure is given here for non-linear systems with dominant linear friction (or anti-friction). For the damped harmonic oscillator, this method yields an exact invariant, as well as a Lagrangian and Hamiltonian, in simple closed forms. In damped (anti-damped). non-linear systems, a series solution for an invariant !and Lagrangian and Hamiltonian) can be obtained by quadratures. This latter treatment is applied to the damped Duffing and van der Pol equations. (C) 1997 Elsevier Science Ltd.