# ANALYTICAL MECHANICS WITH EXERCISES

Degree course:
Corso di First cycle degree in Physics
Academic year when starting the degree:
2019/2020
Year:
2
Academic year in which the course will be held:
2020/2021
Course type:
Supplementary compulsory subjects
Credits:
8
Period:
First Semester
Standard lectures hours:
72
Detail of lecture’s hours:
Lesson (48 hours), Exercise (24 hours)
Requirements:

Knowledge of newtonian mechanics of a single particle and systems of particles, calculus for functions of one and several variables, linear algebra.

Final Examination:
Orale

The final examination consists in a written test (2h), where the student is required to prove a methodological rigor as well as the ability to apply the techniques of analytic mechanics to specific systems. To pass the exam both the problems must be discussed adequately. Apart from particular cases there is no oral exam.

Assessment:
Voto Finale

Teaching goals
The goal is to provide a detailed knowledge of the foundations of lagrangian and hamiltonian mechanics, to discuss physically relevant applications and to illustrate the bases of Hamilton-Jacobi theory.
In the structure of first level degree, analytic mechanics, besides its intrinsic importance, plays also an essential role in the subsequent introduction to quantum mechanics.

Expected learning levels
By the end of the course the student will be able to:
Have a working knowledge of lagrangian and Hamiltonian mechanics
Approach a mechanical problem by finding a suitable set of generalized coordinates, write the corresponding equations of motion, finding the eventual presence of constants of motion and using both qualitative and quantitative techniques to study the main features of motion.

Review of fundamental concepts and Lagrangian form of the equations of motion. (Newtonian mechanics, cardinal equations, D’Alembert principle, generalized coordinates, lagrangian Noether’s theorem, generalized coordinates transformations, generalized potential for a charged particle in an electromagnetic field, the variational Hamilton principle)
Central forces. Kinematics and dynamics of rigid bodies. Oscillations. (central forces and effective potential, Kepler problem, Rutherford scattering, translational and rotational motion of rigid bodies, Euler’s angles, gyroscopes: free motion and the presence of gravity, small oscillations, normal modes of the CO2 molecule)
Hamiltonian mechanics, canonical transformations. (generalized momenta and Hamilton equations, generalized Hamilton principle, relationships between generalized momenta and velocities, canonical transformations: generating functions and Poisson brackets, infinitesimal canonical transformations and conservation laws, lagrangian points, planar stability of L4)
Hamilton-Jacobi theory. (Hamilton-Jacobi equation, complete integrals, separable systems, double centers)

L.D. Landau and E.M. Lifshits, Mechanics
H. Goldstein, C. Poole and J. Safko, Classical Mechanics
J.H. Lowenstein, Essentials of hamiltonian dynamics.
C. Lanczos, The variational principles of mechanics

The course include both theory lectures and recitation sessions, where we will discuss further theoretical developments as well as applications of the theory.

More detailed informations can be obtained from my web page http://www.dfm.uninsubria.it/artuso/Roberto_web_page/Teaching.html, including a pdf copy and recordings of the lectures.
In any case the student may contact me by email Roberto.artuso@uninsubria.it