In variational calculus, the minimality of a given functional under arbitrary deformations with f... more In variational calculus, the minimality of a given functional under arbitrary deformations with fixed end-points is established through an analysis of the so called second variation. In this paper, the argument is examined in the context of constrained variational calculus, assuming piecewise differentiable extremals, commonly referred to as extremaloids. The approach relies on the existence of a fully covariant representation of the second variation of the action functional, based on a family of local gauge transformations of the original Lagrangian and on a set of scalar attributes of the extremaloid, called the corners' strengths [16]. In dis- cussing the positivity of the second variation, a relevant role is played by the Jacobi fields, defined as infinitesimal generators of 1-parameter groups of diffeomorphisms preserving the extremaloids. Along a piecewise differentiable extremal, these fields are generally discontinuous across the corners. A thorough analysis of this poin...
International Journal of Geometric Methods in Modern Physics, 2011
A gauge-invariant formulation of constrained variational calculus, based on the introduction of t... more A gauge-invariant formulation of constrained variational calculus, based on the introduction of the bundle of affine scalars over the configuration manifold, is presented. In the resulting setup, the "Lagrangian" ℒ is replaced by a section of a suitable principal fiber bundle over the velocity space. A geometric rephrasement of Pontryagin's maximum principle, showing the equivalence between a constrained variational problem in the state space and a canonically associated free one in a higher affine bundle, is proved.
Rendiconti del Seminario Matematico della Università di Padova, 1987
L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova... more L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Summary A detailed investigation is performed of reflection and refraction at a plane interface b... more Summary A detailed investigation is performed of reflection and refraction at a plane interface between an elastic body and a viscoelastic one. The incident wave in the elastic body is supposed to be homogeneous while the transmitted waves are necessarily ...
The problem of stability for dynamical systems whose Lagrangian function depends on the derivativ... more The problem of stability for dynamical systems whose Lagrangian function depends on the derivatives of a higher order than one is studied. The difficulty of this analysis arises from the indefiniteness of the Hamiltonian, so that the well-known Lagrange-Dirichlet theorem cannot be used and the methods of the canonical perturbation theory (KAM theory) must be employed. We show, with an
In variational calculus, the minimality of a given functional under arbitrary deformations with f... more In variational calculus, the minimality of a given functional under arbitrary deformations with fixed end-points is established through an analysis of the so called second variation. In this paper, the argument is examined in the context of constrained variational calculus, assuming piecewise differentiable extremals, commonly referred to as extremaloids. The approach relies on the existence of a fully covariant representation of the second variation of the action functional, based on a family of local gauge transformations of the original Lagrangian and on a set of scalar attributes of the extremaloid, called the corners' strengths [16]. In dis- cussing the positivity of the second variation, a relevant role is played by the Jacobi fields, defined as infinitesimal generators of 1-parameter groups of diffeomorphisms preserving the extremaloids. Along a piecewise differentiable extremal, these fields are generally discontinuous across the corners. A thorough analysis of this poin...
International Journal of Geometric Methods in Modern Physics, 2011
A gauge-invariant formulation of constrained variational calculus, based on the introduction of t... more A gauge-invariant formulation of constrained variational calculus, based on the introduction of the bundle of affine scalars over the configuration manifold, is presented. In the resulting setup, the "Lagrangian" ℒ is replaced by a section of a suitable principal fiber bundle over the velocity space. A geometric rephrasement of Pontryagin's maximum principle, showing the equivalence between a constrained variational problem in the state space and a canonically associated free one in a higher affine bundle, is proved.
Rendiconti del Seminario Matematico della Università di Padova, 1987
L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova... more L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Summary A detailed investigation is performed of reflection and refraction at a plane interface b... more Summary A detailed investigation is performed of reflection and refraction at a plane interface between an elastic body and a viscoelastic one. The incident wave in the elastic body is supposed to be homogeneous while the transmitted waves are necessarily ...
The problem of stability for dynamical systems whose Lagrangian function depends on the derivativ... more The problem of stability for dynamical systems whose Lagrangian function depends on the derivatives of a higher order than one is studied. The difficulty of this analysis arises from the indefiniteness of the Hamiltonian, so that the well-known Lagrange-Dirichlet theorem cannot be used and the methods of the canonical perturbation theory (KAM theory) must be employed. We show, with an
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