Mechanical response and the initial value problem

Abstract

Solutions to canonical mechanical equations are considered as functionals of a source function σ, which parametrizes the source. The coefficients of the functional Taylor series expansions in σ for the solutions are constructed from an infinite sequence of time ordered Poisson brackets, which in turn can be expressed in terms of the derivatives of the solution to the homogeneous equation with respect to its initial canonical momenta. Thus the functional Taylor series expansion is formally defined once the initial value problem for the possibly nonlinear homogeneous equation has been solved. A time ordered exponential operator acting on the homogeneous solution, which may be viewed as a ’’response’’ extension of the Lie series operation, simply represents the series expansion.