  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.
