60/64/185 - MECHANICS 2
Academic Year 2022/2023
Free text for the University
FRANCESCO DEMONTIS (Tit.)
- Teaching style
- Lingua Insegnamento
|[60/64] MATHEMATICS||[64/00 - Ord. 2017] PERCORSO COMUNE||8||64|
1. Acquiring knowledge and understanding.
The course is devoted to students in the third year of the Bachelor's degree in Mathematics. It aims at providing a working knowledge of the main facts of analytic mechanics and, in particular, the students should be able to apply the lagrangian and hamiltonian formalism to study holomic systems with a finite number of freedom degree (for example, composed by a finite number of rigid bodies). These topics are presented by providing a rigorous theoretical justifications.
2. Applying knowledge and understanding.
Possible applications of the methods introduced during the lectures will be discussed both for the solution of the mathematical problems and for the solution of typoical problems arising in other disciplines such as physics or engineering.
3. Making informed judgements and choices.
This course allows assiduous students to achieve adeguate knowledge for applying the techniques studied to the solution of mathematical problems which typically can be encountered in Mechanics.
4. Communicating knowledge and understanding.
The evaluation of the written test takes into account the ability of the student to give a methodical and consistent exposition of the topics of the program necessary to find the solutions of the problems assigned. The ability of communicating is further evaluated during the oral interview.
5. Abilities to continue learning.
This course allows assiduous students to acquire a basic expertise which is sufficient to understand advanced mathematical texts on Analytical Mechanics.
It is necessary that the student has already passed Mechanics 1. The course requires a good knowledge of multivalued calculus (partial derivatives, differentiability, studying maxima and minima, multiple integrals, differential equations) and also of curves and surfaces, either in parametric form or as a set of zeros of one or two functions.
1. Newton equation (cardinal equations). D'Alembert's principle and Lagrange's equation. Determinism of the Lagrange equations. The configuration space of a holonomic mechanical system. Generalized momentum. First integrals and cyclic coordinates. Conservation theorems and symmetry properties. Weierstrass' discussion and problems regarding holonomic systems with one degree of freedom. Holonomic systems with two degree of freedom: exercises. Noether's theorem.
2. The calculus of variations. The variational approach to mechanics; Hamilton's principle. Some techniques of the calculus of variations (Fundamental lemma of calculus of variations). Derivation of Lagrange's equations from Hamilton's principle.
3. The Hamilton Equations of Motion. Legendre transformations and the Hamilton equations of motion. Cyclic coordinates and Routh's procedure. Conservation theorems and the physical meaning of the Hamiltonian: Examples.
4. Canonical Transformation: definition. The equations of canonical transformation. Examples of canonical transformations. The integral invariants of Poincarè. Lagrange and Poisson brackets as canonical invariants. The equations of motion in Poisson bracket notation. Simplectic formulation.
5. Equilibrium Positions and Stability. Definition of equilibrium positions for a mechanical system and, in particular, for a holonomic system. Stable and unstable configurations of equilibrium. The Lagrange-Dirichlet theorem (for stable positions of equilibrium) and Liapunov 's theorem (for unstable positions of equilibrium).
6. Small Oscillations. Small oscillations for conservative systems with one or two degrees of freedom about a configuration of stable equilibrium. General case: The eigenvalue equation and the principal axis transformation. Frequencies of free vibration and normal coordinates.
7. Two-body problem. Introduction to the problem. Reduction to the case of one point under a central force (the so-called reduced two bodies problem). First integrals in the reduced two bodies problem.
Simple special cases of the three body problem.
8. Dynamics of rigid bodies. Ruler equations for a rigid body with a fixed point.
The course consists of 64 lecture hours including problem sessions.
In order to make the teaching as much efficient as possible, the theoretical topics are supported by the discussion of exercises given during the lectures and given by the teacher ( 10-15 hours the so-called "tutoraggio"). The teacher offers constant assistance to the students during the whole year both by personal interviews and by means of e-mail messages. The main tools to support teaching is a MS Team channel built for the course and made available a few days before the beginning of the lectures. In this channel the student will find information updated in real time including: a diary reporting the topics treated in each lecture, information on teaching activities and additional documents to support learning.
Verification of learning
The evaluation of the students is carried out by means of a written exam and an oral exam. The written exam includes exercises which require operative skills and knowledge of the theory on the entire contents of the course. Only those students who have at least a score of 18/30 in the written exam will be admitted to the oral exam. The oral exam includes a discussion of the exercises of the written exam where some difficulties have been detected, and some other questions to verify the global understanding of the program.
To pass the exam, the student should show to have acquired a sufficient knowledge of all of the topics of the course. To obtain the maximal grade (30-30 with "cum laude"), the student should instead show to have acquired an excellent knowledge of all of the topics of the course.
Herbert GOLDSTEIN, Charles POOLE, John SAFKO, Meccanica Classica, Zanichelli – Bologna (2005). Exists as: Classical Mechanics
Books to be consulted:
a) J.R. Taylor, Classical Mechanics, Pergamon Press, Oxford-Toronto (2006);
b) T. Levi Civita e U. Amaldi, Lezioni di meccanica razionale e complementi alle lezioni di meccanica razionale, Volumi 1 e 2 curato da Maschio G., Cirillo N. M., Ruggeri T. Editore: Compomat; Published January 2013 (reprinted from the well-known text published by Zanichelli);
c) A. Fasano, S. Marmi, Analytical Mechanics, Oxford Graduate Texts, Oxford University Press, Oxford, 2002.
d) F. Talamucci, Esercizi Svolti sul Formalismo Lagrangiano e Hamiltoniano, Edizioni Nuova Cultura, Roma, 2014.
The lectures may be followed using Microsoft Teams [team: Meccanica 2, code: pzxzmzg]. Appointments can be made by email: firstname.lastname@example.org