Internal Boundary Variations and Discontinuous Transversality Conditions in Mechanics

The aim of the analysis is to recover the impact equation and the jump in the total energy of a Lagrangian system over an impact from the stationarity conditions of a modified action integral. The analysis is accomplished by introducing internal boundary variations and thereby obtaining discontinuous transversality conditions as the stationarity conditions of the impulsive action integral. An impact in mechanics is defined as a discontinuity in the generalized velocities of a mechanical system which is induced by some impulsive forces. An interaction with some constraints may result in an impact and give rise to impulsive forces. The instant of impulsive action where a discontinuity in the generalized velocities occurs is considered as an internal boundary in the time domain. The consideration of certain types of variations at the internal boundaries, which are called internal boundary variations by the author, give rise to discontinuous transversality conditions. By introducing a boundary at an instant of a discontinuity, one has to notice that it has a bilateral character, in the sense that the boundary constitutes an upper boundary for one segment of the interval, whereas for the other segment it constitutes a lower boundary in the time domain. The constraints are therefore introduced symmetrically with respect to pre-impact and post-impact states. It is shown that the impact equation and the energy balance over an impact can be obtained in the form of stationarity conditions for the general impact case by applying the discontinuous transversality conditions. The stationarity conditions are obtained by the application of subdifferential calculus techniques to a suitable extended-valued lower semi-continuous generalized Bolza functional, which in this case is the impulsive action integral, that is evaluated on multiple intervals.