Slow dynamics via degenerate variational asymptotics

We introduce the method of degenerate variational asymptotics for a class of singularly perturbed ordinary differential equations whose lead- ing order behavior is dominated by gyroscopic forces. Such systems exhibit dynamics on two separate time scales which are dynamically linked with no explicit splitting into slow and fast subsystems. We derive approximate equa- tions for the slow motion to arbitrary order by performing an asymptotic ex- pansion of the Lagrangian rather than the Euler-Lagrange equations of motion themselves. Rigorous justification of the method is provided in two different settings. For harmonic potentials, we show that the method can be understood explic- itly in terms of perturbation theory for finite dimensional linear eigenvalue problems. In the general case, we resort to an indirect analysis involving a nonvariational auxiliary model. We illustrate our analytical results by numer- ical simulation.