Università Roma Tre

Newton's equations. Positional and conservative forces. Conservation of mechanical energy.

Equilibria and stability. Linearization around an equilibrium. Small oscillations.

Mechanical systems with one degree of freedom: one-dimensional conservative motions.
Systems with more degrees of freedom. Central forces. The Kepler problem. Kepler's laws.

Lagrangian mechanics: Lagrangian action, stationarity principle, Euler-Lagrange equations. Least action principle.
Constrained motions: olonomic constraints, regular constraints, independent constraints.
The D'Alembert principle. Stationarity principle for mechanical systems with ideal constraints. Generalized energy. Cyclic variables. Routh's reduction method. Noether's theorem.

Change of reference system. Transformation laws for coordinates, velocities and accelerations/forces. Rotation matrix and angular velocity. Fictitious forces: inertial forces, Coriolis' force, centrifugal force.
Rigid body: cinematics and dynamics. The matrix of inertia: principal axes and momenta. Koenig's theorem. Euler's equations.

Introduction to Hamiltonian mechanics: Legendre's transform, Hamiltonian function. Hamilton's equations. Liouville's theorem. Hamilton's and Maupertuis' variational principles. Symplectic matrices. Canonical transformations. Poisson's brackets. Fundamental commutation relations. Generating functions. Hamilton-Jacobi equation. Action-angle variables. Canonical integrability.

- G. Gallavotti: The elements of mechanics, Springer-Verlag, 1983, available online on:
http://ricerca.mat.uniroma3.it/ipparco/pagine/libri.html]

- L.D. Landau, E.M. Lifshitz: Mechanics, Butterworth-Heinemann, 1976.

- V.I. Arnol’d: Mathematical Methods of Classical Mechanics, Springer Graduate Texts in Mathematics.

- G. Gentile: Introduzione ai Sistemi Dinamici: 1 e 2, available online on:
http://www.mat.uniroma3.it/users/gentile/2014-2015/FM410/testo.html