Averaging methods for force controlled and acceleration controlled Lagrangian systems

Recent research has shown that for the class of controlled Lagrangian systems having fewer control inputs than configuration variables, one may blur the distinction between directly controlled states and the corresponding input variables in analyzing the response to oscillatory forcing. Following this approach, stable responses are associated with local minima of an energy-like quantity which we have called the averaged potential. Construction of the averaged potential involves first constructing a reduced Lagrangian to which a Hamiltonian is associated by means of a restricted Legendre transformation. The Hamiltonian is time varying, but by simple averaging one obtains a canonical averaged Hamiltonian from which the averaged potential is immediately determined. Also possible is an averaging analysis of the full (unreduced) system under high-frequency oscillatory forcing. Under suitable symmetry conditions, the averaged effect of an oscillatory input may also be studied in terms of a certain averaged potential which in general differs from the one obtained for the reduced system. In the present paper we discuss the differences between these two approaches and the resulting averaged potentials.