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Geometric constrained variational calculus. III: The second variation (part II). (English) Zbl 1345.49026

Summary: 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 [E. Massa et al., Int. J. Geom. Methods Mod. Phys. 13, No. 1, Article ID 1550132, 31 p. (2016; 06541919)], is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both necessary and sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the extensibility 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.
Editorial remark: for part II, see [loc. cit.], for part I, see [E. Massa et al., Int. J. Geom. Methods Mod. Phys. 12, No. 5, Article ID 1550061, 42 p. (2015; Zbl 1319.49031)].

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
93B27 Geometric methods
49J99 Existence theories in calculus of variations and optimal control
70F25 Nonholonomic systems related to the dynamics of a system of particles

Citations:

Zbl 1319.49031
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References:

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