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Cubic polynomials and optimal control on compact Lie groups
Abrunheiro, L.; Camarinha, M.; Clemente-Gallardo, J.This paper analyzes the Riemannian cubic polynomials’s problem from a Hamiltonian point of view. The description of the problem on compact Lie groups is particulary explored. The state space of the second order optimal control problem considered is the tangent bundle of the Lie group which also has a group structure. The dynamics of the problem is described by a presymplectic formalism associated with the canon...
Jacobi manifolds, Dirac structures and Nijenhuis operators
Clemente-Gallardo, J.; Costa, J. M. Nunes daIn a recent paper [2], we studied the concept of Dirac-Nijenhuis structures. We de ned them as deformations of the canonical Lie algebroid structure of a Dirac bundle D de ned in the double of a Lie bialgebroid (A;A¤) which satisfy certain properties. In this paper, we introduce the concept of generalized Dirac- Nijenhuis structures as the natural analogue when we replace the double of the Lie bialgebroid by th...
Dirac-Nijenhuis structures
Clemente-Gallardo, J.; Costa, J. M. Nunes daWe introduce the concept of Dirac-Nijenhuis structures as those manifolds carrying a Dirac structure and admitting a deformation by Nijenhuis operators which is compatible with it. This concept generalizes the notion of Poisson-Nijenhuis structure and can be adapted to include the Jacobi-Nijenhuis case.
Dirac structures for generalized Lie bialgebroids
Costa, M. Nunes da; Clemente-Gallardo, J.We introduce the notion of Dirac structure for a generalized Courant algebroid. We show that the double of a generalized Lie bialgebroid is a generalized Courant algebroid. We present some examples and we obtain, as a particular case of our definition, the notion of E1(M)-Dirac structure introduced by Wade.
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