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First Semester 
Teaching style
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/64]  MATHEMATICS [64/00 - Ord. 2012]  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 on the main facts of analytic mechanics and, in particular, the students should be able to apply the lagrangian and hamiltonian formalism to study a mechanical system composed of a finite number of rigid bodies. These topics are presented by providing a rigorous theoretical justification.
2. Applying knowledge and understanding.
Possible applications of the methods treated during the course will be discussed, both for the solution of mathematical problems and for the solution of problems arising in in other disciplines such as Physics or Engeeniring.
3. Making informed judgements and choices.
This course allows assiduous students to achieve 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 assigned exercises. The communicating knowledge is further analyzed 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 for widening autonomously their knowledge in 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, maximum and minimum study, integrals, differential equations) and also of curves and surfaces.


1. Lagrange's equations. D'Alambert's principle and Lagrange's equation. Determinism of the Lagrange's equations. The configuration space of an holonomic mechanical system. Generalized momentum. First integral and ignorable variables. The law of conservation of energy. Conservation theorems and symmetries properties. The elimination of ignorable variables. Holonomic systems with one degree of freedom: exercises. Holonomic systems with two degree of freedom: exercises. Spontaneous motion and geodetics.
2. Two bodies 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.
3. 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. Jacobi's principle; the principle of least action.
4. 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.
5. Canonical Transformation. Definition. Sufficient condition ensuring the canonicity of a given transformation (Lie's condition of canonicity). 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.
6. Equilibrium Positions and Stability. Definition of equilibrium positions for a mechanical system and, in particular, for an holonomic system. Stable and unstable configurations of equilibrium. The Lagrange-Dirichlet's theorem (for stable positions of equilibrium) and the Liapunov 's theorem (for unstable positions of equilibrium).
7. Small Oscillations. Small oscillations for conservative systems with one or two degree of freedom around a configuration of stable equilibrium. General case: The eigenvalue equation and the principal axis transformation. Frequencies of free vibration and normal coordinates.
8. Dynamic of rigid bodies. Newton equations. Motion of a rigid body with a fixed axis. Compounded pendulum. Huygens's Theorem. Euler's dinamyc equations. Motion of a rigid body around to its center of mass. Rigid bodies with giroscopic structure with respect to one of their point. Euler's equation for a rigid body with giroscopic sturcture. Inertial precession and their equation of motion. First integral. Brief notes on the integration of the equation of motion for the inertial precession. Permanent rotations.

Teaching Methods

The course consists of 64 lecture hours. Lectures will be given by using either chalk and blackboard or slides. In order to make the teaching as efficient as possible, the theoretical topics are immediately accompanied by exercises and solutions of grading written tests. Furthermore, the teacher will lead 15-20 hours of tutorial activity to assist the students while they study for the final grading. 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 are the teacher's personal web site http://people.unica.it/francescodemontis/
It provides information updated in real time, including: a diary reporting the topics treated in each lecture, information on teaching activities, additional documents to support learning, texts and solutions of grading tests.

Verification of learning

Depending on the evolution of the emergency due to COVID-19, the exam could be carried out by using the software Teams.

The assessment of learning of the students is assessed by means of a written exam and an oral exam. The written exam includes exercises which requires operative skills and knowledge of the theory on the entire contents of the course to be solved. Only those students who have scored 18/30 at least in the written exam will be admitted to the oral exam. The oral exam includes, eventually, a discussion on 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)

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