On the Herglotz variational problem

Abstract

A geometric approach to the Herglotz problem is developed, based on the bundle of affine scalars on the configuration manifold of the given system. The environment, originally introduced to formalize the gauge structure of Lagrangian Mechanics [E. Massa, E. Pagani, and P. Lorenzoni, Transp. Theory Stat. Phys. 29, 69 (2000)], provides the natural setting for the representation of the Herglotz functional as well as for the study of its extremals. Various aspects of the problem are considered: the Lagrangian approach, leading to a generalization of the Poincaré-Cartan algorithm; the parametric approach, involving the introduction of an appropriate super-Lagrangian; the corresponding Hamiltonian and super-Hamiltonian counterparts; the relationship between the Herglotz problem and a constrained variational problem; the evaluation of the abnormality index [Massa et al., Int. J. Geom. Methods Mod. Phys. 12, 1550061 (2015)] of the resulting extremals; the gauge structure of the theory and the consequent existence of Herglotz’s functionals gauge-equivalent to ordinary action functionals.