Mathematical Methods of Analytical Mechanics

The book's calculations use tensor geometry and geometry of variation calculation. Invariance properties are associated with Noether's theorem. The methods of integration, as Jacobi's method, are deduced. The Maupertuis principle corresponding to the conservation of energy of material systems leads to quantum mechanics. We deduce the various spaces underlying the analytical mechanics which lead to the Poisson algebra and the symplectic geometry.

The aim of this book is to:

• To study the calculations of the geometry of the tensor and the geometry of the calculation of the variation
• Understanding the Maupertuis principle that corresponds to the energy conservation of material systems
• Define the invariance properties associated with the Noether theorem
• Talking about phase space, Liouville's theorem
• Identify small movements and different types of stabilities

Applied mathematicians and physicists who wish to obtain a rapid knowledge of the basics of analytical mechanics under a very geometric aspect. Physicists who are involved in statistical mechanics and use classical theorems of the space of phases in mechanics

Part 1. Introduction to the variation calculus
1. The elementary methods of variation calculus
2. Variation of a curvilinear integral
3. Noether's Theorem

Part 2. Applications to the analytical mechanics
4. The methods of analytical mechanics
5. Integration method of Jacobi
6. Spaces of mechanics - Poisson's brackets

Part 3. Properties of mechanical systems
7. Properties of the phase-space
8. Oscillations and small motions of mechanical systems
9. Stability of periodical systems