Lagrangian equations with reciprocal time scales and their application to dissipative systems

A new variational principle, suitable for application to dissipative systems, is devised by considering Lagrangian functions which depend on a set of generalized coordinates q(t) and q(T-t) with reciprocal time scales t in (O, T) and t*=T-t, and their time-derivatives up to the second order. Corresponding Lagrangian equations are derived by the classical variational procedure, and a specific Lagrangian is designed for linear dissipative systems. This Lagrangian is then tested by the Rayleigh-Ritz method, which provides very accurate approximate solutions of a two-point boundary value problem and an initial value problem for a linear dissipative oscillator.