Modeling of inelastic impacts with the help of smooth-functions

There exist two well-known essentially non-linear models for approximate description of the elastic impacts with the help of smooth functions: even-power potential for two-sided impact and inverse-power potential for one-sided impact. It is demonstrated that both models may be generalized to describe the case of inelastic impact with velocity-independent recovery coefficient less than unity. The modification is achieved by adding another strongly non-linear dissipative term to each of the model equations. Condition of velocity independence leads to unique choice of this term for each model; both resulting equations turn out to be completely integrable despite non-linearity and lack of energy conservation. The uniqueness and integrability are proved by means of Lie infinitesimal generators. Use of modified models for problems in higher dimensions is demonstrated by numeric examples. The results may be generalized for complex potentials used in dynamic models of energy pumping.