Geometrical structures on the cotangent bundle

Abstract

In this paper we study the geometrical structures on the cotangent bundle using the notions of adapted tangent structure and regular vector fields. We prove that the dynamical covariant derivative on $T^{*}M$ fix a nonlinear connection for a given $\mathcal{J}$-regular vector field. Using the Legendre transformation induced by a regular Hamiltonian, we show that a semi-Hamiltonian vector field on $T^{*}M$ corresponds to a semispray on $TM$ if and only if the nonlinear connection on $TM$ is just the canonical nonlinear connection induced by the regular Lagrangian.Comment: International Journal of Geometric Methods in Modern Physics, 201

Симметрии Гамильтоновых Систем На Алгеброидах Ли

“…Для заданного J -регулярного сечения ρ на T E * производная Ли L ρ определяет производную тензора на T E * , но не сохраняет некоторые геометрические структуры, например, адаптированную касательную структуру или нелинейную связность. Далее, используя нелинейную связность, введем производную тензора на T E * , называемую динамической ковариантной производной, которая сохраняет некоторые геометрические структуры (см., например, [54,57] для случая кокасательного расслоения).…”

“…Эти геометрические структуры приводят к понятиям эндоморфизма Якоби и динамической ковариантной производной на T * M (см. [54,57]), которые можно использовать для поиска инвариантных уравнений инфинитезимальных симметрий для гамильтоновых систем (см. [56]).…”

Symmetries and conservation laws of Hamiltonian systems

“…However, the existence of a pseudo-metric structure or a regular Hamiltonian on T * M permit us to define an adapted tangent structure J and a regular vector field ρ which induce a nonlinear connection [24]. These geometrical structures permit us to introduce the Jacobi endomorphism and dynamical covariant derivative on T * M (see [26], [27]) which will be used in this paper in order to find the invariant equations of the infinitesimal symmetries of Hamiltonian systems. In fact, this work containts the ideas proposed by the author in [27].…”

“…These geometrical structures permit us to introduce the Jacobi endomorphism and dynamical covariant derivative on T * M (see [26], [27]) which will be used in this paper in order to find the invariant equations of the infinitesimal symmetries of Hamiltonian systems. In fact, this work containts the ideas proposed by the author in [27]. Different types of symmetries and conservation laws for Hamiltonian systems can be found, for example, in [3], [9], [15], [18], [23], [30].…”

“…The paper is organized as follows. In the second section the preliminary geometric structures on the cotangent bundle are recalled (see for instance [22], [24], [25], [26], [27], [33] and references therein). We introduce the Berwald linear connection D on T * M induced by a nonlinear connection N and study its properties in subsection 2.1.…”