Analysis of Nonholonomicity Value of Some Hamiltonian Fields

In classical mechanics such notion as nonholonomicity is applied only to constraints put on a dynamical system. Besides, Pfaffian nonholonomic constraints might be associated with vector fields. The Nonholonomicity value is one of the principal characteristics of such fields, which determines properties of geometry of these vector fields. However, the application of this characteristic in the geometry of vector fields was restricted only to fields in Euclidean spaces. Some generalization of nonholonomicity value of vector fields in non-Euclidean spaces is proposed in this paper. For this purpose the nonholonomicity value is considered as a trilinear form. It is obvious that the coefficients of this form are connected with the components of the metric tensor of the space, where a vector field is defined. So generalization of metric tensor on non-Euclidean spaces generates the generalization of the coefficients of trilinear form, which in its turn generates the generalization of nonholonomicity value. As an example, the nonholonomicity values of Hamiltonian vector fields in sympletic spaces are analyzed in this article. Also it is important to find out whether a mechanical interpretation of the received results exists and can we actually apply this method to Hamiltonian fields.

nonholonomicity value, nonholonomic constraints, Hamiltonian vector field, integrability of differential forms, Pfaffian form, Frobenius theorem