Dual bundle

Abstract: We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds, whose links with Lie algebroids are very close.

Abstract: In this paper, we study the underlying geometry in the classical Hamilton-Jacobi equation. The proposed formalism is also valid for nonholonomic systems. We first introduce the essential geometric ingredients: a vector bundle, a linear almost Poisson structure and a Hamiltonian function, both on the dual bundle (a Hamiltonian system). From them, it is possible to formulate the Hamilton-Jacobi equation, obtaining as a particular case, the classical theory. The main application in this paper is to nonholonomic mechanical systems. For it, we first construct the linear almost Poisson structure on the dual space of the vector bundle of admissible directions, and then, apply the Hamilton-Jacobi theorem. Another important fact in our paper is the use of the orbit theorem to symplify the Hamilton-Jacobi equation, the introduction of the notion of morphisms preserving the Hamiltonian system; indeed, this concept will be very useful to treat with reduction procedures for systems with symmetries. Several detailed examples are given to illustrate the utility of these new developments.

Abstract: A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present in this paper the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds.

Abstract: 1 Geometry of the k-Tangent Bundle TkM.- 1.1 The Category of k-Accelerations Bundles.- 1.2 Liouville Vector Fields. k-Semisprays.- 1.3 Nonlinear Connections.- 1.4 The Dual Coefficients of a Nonlinear Connection.- 1.5 The Determination of a Nonlinear Connection.- 1.6 d-Tensor Fields. N-Linear Connections.- 1.7 Torsion and Curvature.- 2 Lagrange Spaces of Higher Order.- 2.1 Lagrangians of Order k.- 2.2 Variational Problem.- 2.3 Higher Order Energies.- 2.4 Jacobi-Ostrogradski Momenta.- 2.5 Higher Order Lagrange Spaces.- 2.6 Canonical Metrical N-Connections.- 2.7 Generalized Lagrange Spaces of Order k.- 3 Finsler Spaces of Order k.- 3.1 Spaces F(k)n.- 3.2 Cartan Nonlinear Connection in F(k)n.- 3.3 The Cartan Metrical N-Linear Connection.- 4 The Geometry of the Dual of k-Tangent Bundle.- 4.1 The Dual Bundle (T*k M, ?*k, M).- 4.2 Vertical Distributions. Liouville Vector Fields.- 4.3 The Structures J and J*.- 4.4 Canonical Poisson Structures on T*kM.- 4.5 Homogeneity.- 5 The Variational Problem for the Hamiltonians of Order k.- 5.1 The Hamilton-Jacobi Equations.- 5.2 Zermelo Conditions.- 5.3 Higher Order Energies. Conservation of Energy ?k ?1(H).- 5.4 The Jacobi-Ostrogradski Momenta.- 5.5 Noether Type Theorems.- 6 Dual Semispray. Nonlinear Connections.- 6.1 Dual Semispray.- 6.2 Nonlinear Connections.- 6.3 The Dual Coefficients of the Nonlinear Connection N.- 6.4 The Determination of the Nonlinear Connection by a Dual k-Semispray.- 6.5 Lie Brackets. Exterior Differential.- 6.6 The Almost Product Structure ?. The Almost Contact Structure $$ \mathbb{F} $$.- 6.7 The Riemannian Structure G on T*kM.- 6.8 The Riemannian Almost Contact Structure $$(\mathop \mathbb{G}\limits^ \vee ,\mathop \mathbb{F}\limits^ \vee )$$.- 7 Linear Connections on the Manifold T*kM.- 7.1 The Algebra of Distinguished Tensor Fields.- 7.2 N-Linear Connections.- 7.3 The Torsion and Curvature of an N-Linear Connection.- 7.4 The Coefficients of a N-Linear Connection.- 7.5 The h-,??- and ?k-Covariant Derivatives in Local Adapted Basis.- 7.6 Ricci Identities. Local Expressions of d-Tensor of Curvature and Torsion. Bianchi Identities.- 7.7 Parallelism of the Vector Fields on the Manifold T*kM.- 7.8 Structure Equations of a N-Linear Connection.- 8 Hamilton Spaces of Order k ? 1.- 8.1 The Spaces H(k)n.- 8.2 The k-Tangent Structure J and the Adjoint k-Tangent Structure J*.- 8.3 The Canonical Poisson Structure of the Hamilton Space H(k)n.- 8.4 Legendre Mapping Determined by a Lagrange Space L(k)n= (M, L).- 8.5 Legendre Mapping Determined by a Hamilton Space of Order k.- 8.6 The Canonical Nonlinear Connection of the Space H(k)n.- 8.7 Canonical Metrical N-Linear Connection of the Space H(k)n.- 8.8 The Hamilton Space H(k)n of Electrodynamics.- 8.9 The Riemannian Almost Contact Structure Determined by the Hamilton Space H(k)n.- 9 Subspaces in Hamilton Spaces of Order k.- 9.1 Submanifolds $${T^{*k}}\mathop M\limits^ \vee$$ in the Manifold T*kM.- 9.2 Hamilton Subspaces $${ {\mathop H\limits^ \vee} ^{(k)m}}$$ in H(k)n. Darboux Frames.- 9.3 Induced Nonlinear Connection.- 9.4 The Relative Covariant Derivative.- 9.5 The Gauss-Weingarten Formula.- 9.6 The Gauss-Codazzi Equations.- 10 The Cartan Spaces of Order k as Dual of Finsler Spaces of Order k.- 10.1 C(k)n-Spaces.- 10.2 Geometrical Properties of the Cartan Spaces of Order k.- 10.3 Canonical Presymplectic Structures, Variational Problem of the Space C(kn).- 10.4 The Cartan Spaces C(k)n as Dual of Finsler Spaces F(k)n.- 10.5 Canonical Nonlinear Connection. N-Linear Connections.- 10.6 Parallelism of Vector Fields in Cartan Space C(kn).- 10.7 Structure Equations of Metrical Canonical N-Connection.- 10.8 Riemannian Almost Contact Structure of the Space C(kn).- 11 Generalized Hamilton and Cartan Spaces of Order k. Applications to Hamiltonian Relativistic Optics.- 11.1 The Space GH(kn).- 11.2 Metrical N-Linear Connections.- 11.3 Hamiltonian Relativistic Optics.- 11.4 The Metrical Almost Contact Structure of the Space GH(kn).- 11.5 Generalized Cartan Space of Order k.- References.