The problem of minimality for constrained variational calculus is analyzed within the class of piecewise differentiable extremaloids. A fully covariant representation of the second variation of the action functional based on a family of local gauge transformations of the original Lagrangian is proposed. The necessity of pursuing a local adaptation process, rather than the global one described in [1] is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both the necessary and the sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the prolongability of the Jacobi fields across the corners. Eventually, in the appendix, an alternative approach to the concept of strength of a corner, more closely related to Pontryagin’s maximum principle, is presented.

Geometric constrained variational calculus. III: The second variation (Part II) / Massa, Enrico; Luria, Gianvittorio; Pagani, Enrico. - In: INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS. - ISSN 0219-8878. - STAMPA. - 13:4(2016), p. 1650038. [10.1142/S0219887816500389]

Geometric constrained variational calculus. III: The second variation (Part II)

Pagani, Enrico
2016-01-01

Abstract

The problem of minimality for constrained variational calculus is analyzed within the class of piecewise differentiable extremaloids. A fully covariant representation of the second variation of the action functional based on a family of local gauge transformations of the original Lagrangian is proposed. The necessity of pursuing a local adaptation process, rather than the global one described in [1] is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both the necessary and the sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the prolongability of the Jacobi fields across the corners. Eventually, in the appendix, an alternative approach to the concept of strength of a corner, more closely related to Pontryagin’s maximum principle, is presented.
2016
4
Massa, Enrico; Luria, Gianvittorio; Pagani, Enrico
Geometric constrained variational calculus. III: The second variation (Part II) / Massa, Enrico; Luria, Gianvittorio; Pagani, Enrico. - In: INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS. - ISSN 0219-8878. - STAMPA. - 13:4(2016), p. 1650038. [10.1142/S0219887816500389]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/164395
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