Dynamical systems of Lagrangian and Hamiltonian mechanical systems

Abstract

In Part I of this paper the dynamical systems of the Lagrangian mechanical system $\Sigma_L = (M, L(x,y),\ F_e(x,y))$ are defined and investigated. In Theorem 3.1 we prove the existence of a canonical dynamical system on the phase space whose integral curves are given by the Lagrange equations of $\Sigma_L$. The particular case of Finslerian mechanical systems is considered. The geometry of $\Sigma_L$ on TM is also described. Part I is a survey of the author's papers [18] [22] [23].

In the Part II for the first time the same problems for the Hamiltonian mechanical systems $\Sigma_H = (M, H(x,p),\ F_e(x,p))$ are studied. In Theorem 10.1, we prove the existence of a canonical dynamical system $\xi$ on the momenta space, whose integral curves are given by the Hamilton equations of $\Sigma_H$. As a particular case the Cartan mechanical systems are examined.