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Resonant forcing of nonlinear systems of differential equations
Full text linkWe study resonances of nonlinear systems of differential equations, including
but not limited to the equations of motion of a particle moving in a potential.
We use the calculus of variations to determine the minimal additive forcing
function that induces a desired terminal response, such as an energy in the
case of a physical system. We include the additional constraint that only
select degrees of freedom be forced, corresponding to a very general class of
problems in which not all of the degrees of freedom in an experimental system
are accessible to forcing. We find that certain Lagrange multipliers take on a
fundamental physical role as the effective forcing experienced by the degrees
of freedom which are not forced directly. Furthermore, we find that the product
of the displacement of nearby trajectories and the effective total forcing
function is a conserved quantity. We demonstrate the efficacy of this
methodology with several examples.Comment: 9 pages, 3 figure
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