CONOSCENZA E COMPRENSIONE:
Al termine del corso lo studente dovrà dimostrare di conoscere i risultati fondamentali del formalismo Lagrangiano e Hamiltoniano. In particolare i legami tra le simmetrie e le costanti del moto e la teoria delle trasformazioni canoniche e simplettiche con l'uso dell' equazione di Hamilton-Jacobi.

CAPACITÀ DI APPLICARE CONOSCENZA:
Al termine del corso lo studente dovrà saper applicare le conoscenze del formalismo Lagrangiano e Hamiltoniano, saper ricavare relazioni tra simmetrie e costanti del moto e saper risolvere sistemi Hamiltoniani usando trasformazioni canoniche e simplettiche.

Newtonian Mechanics

Affine spaces.

Vector fields as dynamical systems. Constant of motions. Qualitative analysis.
Stability of equilibrium solutions.

Mechanical systems, Constraints, Degrees of freedom Lagrangian coordinates.
D'Alembert principle, Cardinal equations of statics and dynamics.
The principle of virtual work. Virtual displacements. Generalised forces.
Lagrange equations, Potential, Lagrangian function, lagrange equations.

Linearisation around an equilibrium position, Small oscillations, Linear
stability, Normal coordinates, resonances.

Geodetics motions.

Variational principles, Euler-Lagrange equations.

Legendre transformation, canonical systems, Hamilton equations,
Constant of motions, Poisson brackets, Symplectic structure, Canonical
transformations.

Hamilton-Jacobi, action-angle variables, integrability.

Rigid body cinematic, Poisson formula, Angular velocity, Euler angles,
Inertia tensor, principal inertia axis, Euler equations in three dimensions.
Procession by inertia, Gyroscopic effects, Lagrangian top.

S. Benenti, Modelli matematici della meccanica, vol. I e II (Celid, 1997).
A. Fasano, S. Marmi, Meccanica analitica (Bollati 1994)
V.I. Arnold, Metodi matematici della meccanica classica (Editori Riuniti,
1979).

Affine spaces.

Vector fields as dynamical systems. Constant of motions. Qualitative analysis.
Stability of equilibrium solutions.

Mechanical systems, Constraints, Degrees of freedom Lagrangian coordinates.
D'Alembert principle, Cardinal equations of statics and dynamics.
The principle of virtual work. Virtual displacements. Generalised forces.
Lagrange equations, Potential, Lagrangian function, lagrange equations.

Linearisation around an equilibrium position, Small oscillations, Linear
stability, Normal coordinates, resonances.

Geodetics motions.

Variational principles, Euler-Lagrange equations.

Legendre transformation, canonical systems, Hamilton equations,
Constant of motions, Poisson brackets, Symplectic structure, Canonical
transformations.

Hamilton-Jacobi, action-angle variables, integrability.

Rigid body cinematic, Poisson formula, Angular velocity, Euler angles,
Inertia tensor, principal inertia axis, Euler equations in three dimensions.
Procession by inertia, Gyroscopic effects, Lagrangian top.

Lecture classes. There will be a tutor who will propose and correct exercises and help students in working groups.

.

A written part on exercises and a oral part for theory.