On Locally Conformally Cosymplectic Hamiltonian Dynamics and Hamilton-Jacobi Theory

Abstract

Cosymplectic geometry has been proven to be a very useful geometric background to describe time-dependent Hamiltonian dynamics. In this work, we address the globalization problem of locally cosymplectic Hamiltonian dynamics that failed to be globally defined. We investigate both the geometry of locally conformally cosymplectic (abbreviated as LCC) manifolds and the Hamiltonian dynamics constructed on such LCC manifolds. Further, we provide a geometric Hamilton-Jacobi theory on this geometric framework.[12]. In the light of these observations, we shall establish Proposition 3.5 determining a Darboux theorem for LCC manifolds.(3) Jacobi Structure of LCC Manifolds. LCS manifolds are Jacobi manifolds, we shall explicitly show the Jacobi character of LCC manifolds in Proposition 4.1.(4) Algebra of One-Forms: Lie Algebroid Realizations. It is known that Jacobi manifolds admit Lie algebroid formalisms [22]. We shall review this for LCS manifolds in Section 2.3 and show that Lichnerowicz-deRham (abbreviated as LdR) exact one-forms constitute a subalgebra.In Section 4.3 we shall write an algebra on the space of one-form sections on LCC. Then we shall discuss the Lie algebroid realization of LCC manifolds.(5) Hamilton-Jacobi Formalism for LCC Hamiltonian Dynamics. Hamilton-Jacobi theory provides a way to solve Hamilton equations. In a recent paper [35], we examined the geometric Hamilton-Jacobi theory for Hamiltonian dynamics on LCS manifolds. Additionally, we took the locally conformal discussions to k-symplectic formalism [34] and jet bundle formalism [33] in order to define locally conformally Hamiltonian field theories. In that work, we provided the locally conformal Hamilton-De Donder-Weyl formalism, as well as the geometric Hamilton-Jacobi theories for these extensions. In this work, we shall present two versions (namely in Theorem 4.4 and Theorem 4.5) of the geometric Hamilton-Jacobi theorem in the realm of LCC Hamiltonian dynamics. These results are a generalization of the HJ theorems obtained for cosymplectic Hamiltonian dynamics in [28].The content. In the following section we summarize Hamiltonian dynamics on symplectic and LCS manifolds and their corresponding geometric HJ theorems. In Section 3, we shall recall cosymplectic manifolds and the Hamilton-Jacobi theorem for cosymplectic Hamiltonian dynamics.Then we shall explain the basics on LCC geometry, and it is in this same section where the symplectization problem of LCC manifolds and Darboux coordinates of LCC are established. In Section 4, dynamics on LCC manifolds is explained, Jacobi and Lie algebroid characters of LCC manifolds will be obtained, and finally the Hamilton-Jacobi theorems for LCC Hamiltonian dynamics will be written.Notation. From now on we consider X(M) to be the space of vector fields on M, whereas Γ k (M) is the space of k-form sections on M. L X is the Lie derivative with respect to the vector field X. An arbitrary almost symplectic manifold is represented by the pair (N, ω) while an arbitrary almost cosymplectic manifold is denoted by ...