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 Mechanics1, 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 index2 of the resulting extremals; the gauge structure of the theory and the consequent existence of Herglotz’s functionals gauge-equivalent to ordinary action functionals.

On the Herglotz variational problem / Massa, Enrico; Pagani, Enrico. - In: JOURNAL OF MATHEMATICAL PHYSICS. - ISSN 0022-2488. - ELETTRONICO. - 2023, 64:10(2023), pp. 10290201-10290220. [10.1063/5.0165641]

On the Herglotz variational problem

Pagani, Enrico
2023-01-01

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 Mechanics1, 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 index2 of the resulting extremals; the gauge structure of the theory and the consequent existence of Herglotz’s functionals gauge-equivalent to ordinary action functionals.
2023
10
Massa, Enrico; Pagani, Enrico
On the Herglotz variational problem / Massa, Enrico; Pagani, Enrico. - In: JOURNAL OF MATHEMATICAL PHYSICS. - ISSN 0022-2488. - ELETTRONICO. - 2023, 64:10(2023), pp. 10290201-10290220. [10.1063/5.0165641]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/397570
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