Constrained Lagrangian dissipative contact dynamics

We show that the contact dynamics obtained from the Herglotz variational principle can be described as a constrained nonholonomic or vakonomic ordinary Lagrangian system depending on a dissipative variable with an adequate choice of one constraint. As a consequence, we obtain the dynamics of contact nonholonomic and vakonomic systems as an ordinary variational calculus with constraints on a Lagrangian with a dissipative variable. The variation of the energy and the other dissipative quantities is also obtained, giving the usual results.We acknowledge the financial support from the Spanish Ministerio de Ciencia, Innovación y Universidades, under Project No. PGC2018-098265-B-C33, the MINECO under Grant Nos. MTM2016-76-072-P and PID2019-106715GB-C21, and the “Severo Ochoa Programme for Centres of Excellence in R & D” (Grant No. CEX2019-000904-S). Manuel Laínz acknowledges MICINN and ICMAT for a FPI-Severo Ochoa predoctoral under Contract No. PRE2018-083203. We also thank the referee for his careful reading of the manuscript and his comments that have allowed us to improve some parts of the work