The non-holonomic Herglotz variational problem

The geometric approach to the study of the Herglotz problem developed in Massa and Pagani [J. Math. Phys. 64, 102902 (2023)] is extended to the case in which the evolution of the system is subject to a set of non-holonomic constraints. The original setup is suitably adapted to the case in study. Various aspects of the problem are considered: the direct derivation of the evolution equations; the super-lagrangian approach; the resulting super-Hamiltonian and its relation with Pontryagin's maximum principle; the abnormality index of the extremals; the invariance properties of the theory and the consequent existence of Herglotz Lagrangians gauge equivalent to ordinary ones.