## Learning objectives

To provide the basic theoretical concepts in Lagrangian and Hamiltonian mechanics. To understand the principles leading to the study of macroscopic systems and to discuss the basic concepts in statistical mechanics and the methods to calculate the thermodynamical properties of macroscopic systems at equilibrium, starting from the statistical distribution of microscopic variables in phase space

## Prerequisites

- - -

## Course unit content

Introduction to Analytical Mechanics.

Statistical Mechanics of Microcanonical, Canonical and Gran Canonical Ensembles.

Applications of the classical ensembles.

## Full programme

- Classical Mechanics in an arbitrary reference frame. Constraints, virtual displacements, generalized lagrangian coordinates. The Lagrangian of a physical systems and the Lagrange equations. Symmetries and conservation laws. Noether's theorem. Small oscillations, normal modes. The Legendre transform and the Hamiltonian. Hamilton's equations. Configuration space and phase space. Poisson brackets.

- Variational principles and Lagrange and Hamilton equations. Elements of calculus of variations. Canonical transformations. Elements of perturbation theory. Examples of relevant Lagrangians and Hamiltonians of physical systems: central forces, changed particles in an electromagnetic field. Infinite degrees of freedom: the vibrating string.

- The statistical description of a macroscopic system. Systems with many degrees of freedom and classical mechanics. Average values without dynamics: statistical ensembles and probability measures. Liouville theorem. The problems of the microscopic approach. Temporal averages and the ergodic hypothesis. Recurrence times and macroscopic variables. How and if equilibrium is reached.

-Brief review of thermodynamics: extensive and intensive variables, thermodynamic potentials, Legendre transformations, response functions. Microcanonical distribution. Boltzmann entropy and its properties. Additivity. Microcanonical classical ideal gas. Gibbs paradox. Entropy and information theory: Shannon entropy.

- Canonical distribution. The partition function and the Helmotz free energy. Energy fluctuation in the canonical ensemble. Fluctuation and response. Maxwell distribution. Equipartition. Equivalence between microcanonical e canonical ensembles. Canonical Ideal Gas.

- Gran canonical distribution. Gran canonical partition function and pressure. Chemical potential. Gran canonical Ideal Gas.

## Bibliography

H. Goldstein- C. Poole - J. Safko, Classical Mechanics

L.D. Laundau - E.M. Lifsits, Mechanics

L.D. Laundau - E.M. Lifsits, Statistical Physics

Lecture notes.

K. Huang - Statistical Mechanics

## Teaching methods

Lectures and exercices

## Assessment methods and criteria

Oral and Written examination

## Other information

Support activity: tutor activity during the course, material from web sites on advanced subjects