Lagrangian and Hamiltonian Geometries. Applications to Analytical Mechanics

zbMATH Open1239.53001arXiv1203.4101MaRDI QIDQ3113882

Publication date: 26 January 2012

Abstract: The aim of the present text is twofold: to provide a compendium of Lagrangian and Hamiltonian geometries and to introduce and investigate new analytical Mechanics: Finslerian, Lagrangian and Hamiltonian. The fundamental equations (or evolution equations) of these Mechanics are derived from the variational calculus applied to the integral of action and these can be studied by using the methods of Lagrangian or Hamiltonian geometries. More general, the notions of higher order Lagrange or Hamilton spaces have been introduced and developed by the present author. The applications led to the notions of Lagrangian or Hamiltonian Analytical Mechanics of higher order. For short, in this text we aim to solve some difficult problems: The problem of geometrization of classical non conservative mechanical systems; The foundations of geometrical theory of new mechanics: Finslerian, Lagrangian and Hamiltonian;To determine the evolution equations of the classical mechanical systems for whose external forces depend on the higher order accelerations. My colleagues Professors M. Anastasiei, I. Bucataru, I. Mihai, K. Stepanovici made important remarks and suggestions and Mrs. Carmen Savin prepared an excellent print-form of the hand-written text. Many thanks to all of them.

Full work available at URL: https://arxiv.org/abs/1203.4101

zbMATH Keywords

Lagrangian semispray Finslerian and Hamiltonian mechanical systems

Mathematics Subject Classification ID

Symplectic manifolds (general theory) (53D05) Applications of differential geometry to physics (53Z05) Special Riemannian manifolds (Einstein, Sasakian, etc.) (53C25) Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds (58B20) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry (53-01) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Global differential geometry of Finsler spaces and generalizations (areal metrics) (53C60) Hamiltonian and Lagrangian mechanics (70Hxx)