## 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

Physics 1 and Basic Courses in Mathematics

## Course unit content

Introduction to Analytical Mechanics. Statistical Mechanics of Microcanonical and Canonical 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. Brief review of thermodynamics: extensive and intensive variables, thermodynamic potentials, Legendre transformations, response functions. 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.

- The Microcanonical Ensemble. Boltzmann entropy and its properties. Additivity. Microcanonical classical ideal gas. Gibbs paradox and correct counting. Entropy and information theory: Shannon entropy.

- The Canonical Ensemble. The partition function and the Helmotz free energy.

Energy fluctuation in the canonical ensemble. Equivalence between microcanonical e canonical ensembles.

## Bibliography

H. Goldstein- C. Poole - J. Safko, Meccanica Classica - Zanichelli

L.D. Laundau - E.M. Lifsits, Meccanica - Ed Riuniti

L.D. Laundau - E.M. Lifsits, Fisica Statistica, Editori Riuniti

K. Huang - Statistical Mechanics - Wiley and Sons

## Teaching methods

Oral and pratical lessons

## Assessment methods and criteria

Oral and written exam.

## Other information

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