Structure and classification of matrices characterizing jointly Lie-admissible brackets and symplectic-admissible two-forms
In this paper we investigate the structure of invertible matrices which have at the same time the property of Lie-admissibility (i.e. their skew-symmetric part characterizes generalized Poission brackets), and the property of sympletic-admissibility (i.e. the skew-symmetric part of their inverse characterizes a symplectic two-form). According to recent studies, these matrices play a role in the problem of generalizing Hamilton's equations for the inclusion of forces non-derivable from a potential. The main tool underlying the present study is the Darboux theorem, which ensures that the skew-symmetric parts of the matrices under consideration can locally be reduced to a canonical form. The Darboux-reduction of one of the skew-symmetric parts then still leaves open the existence of a set of 2n functions a(b), performing he reduction of the skew-symmetric part of the inverse matrix. It is shown that once these functions are known, the determination of the type of matrices under consideration is reduced to the determination of elements of the symplectic group. In addition, more manageable algebraic relations are obtained by introducing the Cayley-parametrization of this group. These relations exhibit more clearly the number of unknowns involved, but they require an additional assumption. The 2n functions a(b) then are proposed as keystones for a classification of these matrices, and correspondingly an analysis is performed on the role of canonical transformations in this classification. A discussion is given of the special class of solutions for which both skew-symmetric parts are each others inverse, and a few other illustrative examples are consturcted in an appendix.
- Research Organization:
- Harvard Univ., Cambridge, MA
- OSTI ID:
- 6789223
- Journal Information:
- Hadronic J.; (United States), Vol. 2:1
- Country of Publication:
- United States
- Language:
- English
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