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inverse problem of the calculus of variations; Helmholtz conditions; nonholonomic constraints; the nonholonomic variational principle; constraint Euler-Lagrange equations; constraint Helmholtz conditions; constraint Lagrangian; constraint ballistic motion; relativistic particle
The inverse problem of the calculus of variations in a nonholonomic setting is studied. The concept of constraint variationality is introduced on the basis of a recently discovered nonholonomic variational principle. Variational properties of first order mechanical systems with general nonholonomic constraints are studied. It is shown that constraint variationality is equivalent with the existence of a closed representative in the class of 2-forms determining the nonholonomic system. Together with the recently found constraint Helmholtz conditions this result completes basic geometric properties of constraint variational systems. A few examples of constraint variational systems are discussed.
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