LEARNING OUTCOMES

In this course will be dealt with the foundations of both Lagrangian and Hamiltonian analytical mechanics and the theory of stability.

AIMS AND LEARNING OUTCOMES

The aim of the lecture is to present analytical mechanics, both from the Lagrange and the Hamiltonian point of view, and its applications to the solution of mechanical problems.
Starting from Newton's laws and the analysis of constrained systems, the Lagrangian formalism is introduced along with the Euler-Lagrange equations. The existence of (local) solutions for Euler-Lagrange equations is examined in detail, focusing in particular on the structure of the kinetic energy. During the course several physically interesting examples are discussed, both concerning systems of point particles and rigid bodies. Furthermore, stability theory à la Ljapunov for autonomous dynamical systems is developed, including its application to small oscillations around stable equilibrium configurations of a mechanical system.
The transition from the Lagrangian to the Hamiltonian formalism is achieved through the Legendre transform. This allows us to deduce Hamilton equations from Euler-Lagrange equations. The natural symplectic structure that appears manifestly in Hamiltonian mechanics is formalized introducing Poisson brackets, whose properties are analyzed in detail. Therefore, we are naturally led to introduce canonical transformations as those coordinate transformations that preserve the form of the Poisson brackets. Further equivalent characterizations of the class of canonical transformations are derived, including the one based on generating functions. The latter characterization of canonical transformations brings us to Hamilton-Jacobi equations, whose goal is to single out a system of canonical coordinates such that Hamilton equations become trivial.
The last part of the lecture focuses on variational principles, both in the Lagrangian and in the Hamiltonian formalism, with some applications to problems both of geometric and of physical flavor.
Throughout the lecture, weekly exercise classes are offered, whose aim is to prepare the student to independently solve a wide range of problems of mechanical nature using the techniques presented during the lecture.
 

TEACHING METHODS

The lecture consists of taught classes. Part of the classes are of theoretic nature (approximately 48 hours). The purpose of those is to present the theoretic aspects of analytical mechanics, along with some concrete examples. This theoretic part is complemented by weekly exercise classes (approximately 24 hours), whose purpose is to show how to solve concrete analytical mechanics problems using the tools presented in the theoretic part of the lecture.

SYLLABUS/CONTENT

Introduction and basic concepts

• Spacetime in classical physics
• Point particle         (only for the 8 credits course)
• Relative motion         (only for the 8 credits course)
• Euler's law of motion for systems of point partivles and rigid bodies         (only for the 8 credits course)

Analytical mechanics of holonomic systems

• Holonomic systems and ideal constraints
• Euler-Lagrange equations
• Lagrange equation and balance equations
• Integrals of motion in the Lagrangian formalism

Introduction to stability of dynamical systems

• Generalities on ordinary differential equations
• Stability and Ljapunov's theory
• Small oscillations for a mechanical system

Hamiltonian mechanics

• Legendre transformation and Hamilton's equations
• Poisson brackets
• Canonical transformations
• Generating functions
• Transformation law for the Hamiltonian 
• Hamilton-Jacobi equation

Variational principles

• Lagrangian case and Hamiltonian case
• Canonical transformations and covariance of the action functional