# RATIONAL MECHANICS

RATIONAL MECHANICS part 1 (6 credits) | RATIONAL MECHANICS part 2 (6 credits) |

## Learning outcomes of the course unit

Knowledge and understanding: the student must acquire the knowledge of foundations of Classical Mechanics viewed as a branch of Mathematical Physics, with a deep understanding of the basic applications of mathematical methods to the study of physical problems. Moreover, the student must become able to read and understand advanced text of Rational Mechanics and Mathematical Physics.

Applying knowledge and understanding: the student must become able to produce formal proofs of results of Classical Mechanics and Mathematical Physics, and to expose, analyze and solve simple problems of Classical Mechanics with a clear mathematical formulation.

Making judgements: the student must become able to construct, develop and apply theoretical reasoning in the context of Classical Mechanics and Mathematical Physics, with a deep ability to distinguish correct and wrong assumptions and methods.

Communication skills: the student must acquire the correct terminology and language of Classical Mechanics and Mathematical Physics and the ability to expose their results and techniques to an audience, in both cases of qualified and unqualified audience.

Learning skills: the student must become able to autonomously continue the study of Classical Mechanics, Mathematical Physics and in general to complete his preparation in Mathematics or in other scientific fields with an open minded approach, and must become able to gain knowledge from specialized text and journals.

## Prerequisites

Basic calculus of the first year courses; mandatory propedeuticities: Mathematical Analysis 1, Geometry 1A.

## Course contents summary

The course aims at providing the foundations and some applications of Classical Mechanics. Therefore, the first part of the course deals with definitions, properties and computational techniques of the mathematical structures typical of Classical Mechanics; the second part deals with foundations, results and applications of Absolute and Relative Kinematics; the third part deals with foundations, results and applications of Classical Dynamics of particles and of systems with a finite number of degrees of freedom, in particolar of the rigid body, with particolar attention to Lagrangian and Hamiltonian Mechanics.

## Course contents

1 - PREREQUISITES

1.1 Linear Algebra and Geometry

1.2 Analysis in Rn

2 – MATHEMATICAL PRELIMINARIES

2.1 Affine spaces

2.2 Isometries

2.3 Applied Vectors

2.4 Vector Analysis

2.5 Differential curves in E3

3 – KINEMATICS

3.1 Absolute Kinematics

3.2 Relative Kinematics

4 – DYNAMICS OF POINT MASSES

4.1 Power and Work

4.2 Conservative Forces

4.3 Dynamics of free point masses

4.4 Dynamics of constrained point masses

4.5 Constitutive Characterization of constraints

4.6 Applications

5 – SYSTEMS OF MASS POINTS

5.1 Center of mass

5.2 Mechanical quantities

5.3 Power and Work

5.4 Conservative Forces

5.5 Classification of constraints

5.6 Kinetic constraints

5.7 Newton’s laws for systems

6 – RIGID BODY DYNAMICS

6.1 Inertia tensor

6.2 Rigid body’s evolution equations:

7 – LAGRANGIAN MECHANICS

7.1 Holonomic systems, configuration space, motion space

7.2 D’Alembert principle, virtual work principle, equilibria

7.3 Kinetic energy in the lagrangian formalism

7.4 Potential energy

7.5 Conservative Lagrange equations

7.6 Prime integrals

7.7 Stability of equilibria, Dirichlet criterium, small oscillations.

7.8 Dynamics of a rigid body with fixed point: Euler equations.

8- HAMILTONIAN MECHANICS

8.1 Phase Space

8.2 Legendre Transformation

8.3 Hamiltonian and Hamilton canonical equations

## Recommended readings

Notes of the lectures in pdf format are provided to the student. In addition to the shared material, the student can personally deepen some of the topics discussed during the course in the following books:

Mauro Fabrizio “Elementi di Meccanica Classica” – Zanichelli Editore;

Levi Civita, T. and Amaldi, U. (2013) “Lezioni di Meccanica Razionale”-- Ed. Compomat;

Biscari P., Ruggeri T., Saccomandi G., Vianello M. “Meccanica Razionale” (3° edizione) -- Unitext Springer (2015); Goldstein H., Poole C., Safko J. “Meccanica Classica” – Zanichelli Editore.

## Teaching methods

The didactic activities are composed of frontal lessons having theoretical character, alternating with sessions pertaining exercises. Theoretical lessons concerns the formal aspects of Classical Mechanics, with its foundations, main results and limits of applicability. Exercises concerns both theoretical applications of the principles of Classical Mechanics and its computational aspects.

## Assessment methods and criteria

The knowledge will be verified through a written test and an oral exam. The written test consists in an exercise based on open questions about a mechanical system and it lasts 3 hours. The oral exam consists in a discussion of the written test and its solution, and questions about all the arguments of the lessons. The oral exam can be taken only if the written test has sufficient mark. Verification is positively valued if and only if both the written test and the oral exam have sufficient marks. A positive result in the written test is valid only for the verification under way, and its validity cannot be extended to subsequent verifications.

The written test can be passed also through two partial tests that will be fixed at the end of the frontal lessons of the two semesters of the course. A positive result in both partial tests allows a single possibility of taking an oral exam along the year.