The purpose of the book is a short presentation of the geometrical theory of Lagrange and Hamilton spaces of order 1 or of order k ≥1, as well as the definition and investigation of some new Analytical Mechanics of Lagrangian and Hamiltonian type of order k ≥1.
The aim of the present monograph is twofold:
1. to provide a Compendium of Lagrangian and Hamiltonian geometries;
2. to introduce and investigate new analytical Mechanics: Finslerian, Lagrangian and Hamiltonian.
The book is divided in three parts:
I. Lagrange and Hamilton spaces;
II. Lagrange and Hamilton spaces of higherorder;
III. Analytical Mechanics of Lagrangian and Hamiltonian mechanical systems.
For short, in this monograph we aim to solve some difficult problems:
 the problem of geometrization of classical nonconservative 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 higherorder accelerations.
