# MATHEMATICAL PHYSICS I

**Academic Year 2023/2024**- Teacher:

**Giulia PICCITTO**

## Expected Learning Outcomes

## Course Structure

Classroom lessons. Should teaching be carried out
in mixed mode or remotely, it may be necessary to introduce changes
with respect to previous statements, in

line with the programme planned and outlined in the syllabus.

Learning assessment may also be carried out on line, should the conditions require it.

## Required Prerequisites

## Detailed Course Content

**Elements of vector calculus. Applied vectors **

**Kinematic of the point. ** Generality. Space and time.
Smooth curves and Frenet reference. Speed and acceleration of a
material point. Plane motion, circular motion, harmonic motion, helical
motion.

**Kinematic of particle systems.** General information
about the constraints for particle systems. Holonomic, nonholonomic,
fixed, mobile, one-sided, two-sided. Degrees of freedom and Lagrangian
parameters. Rigid body. Degrees of freedom of a rigid system. Euler
angles. Poisson formula. Fundamental formula of the kinematics of the
rigid bodies. Rigid motions translational, rotational, helical,
roto-shifters, polar and precession. Act of motion. Mozzi's Theorem.
Relative kinematics. Theorems of velocities and accelerations
composition. Rigid planar motion. Pure rolling motion.

**Dynamics and statics for a material point. ** Principles of dynamics. Statics. Dynamics and statics in non-inertial frame of reference. Terrestrial mechanics. Weight

**Dynamics and statics of particle systems**. Center of
mass. Moment of inertia. Huygens theorem. Inertia ellipsoide and
Principal axis frame. Examples and applications. Cardinal equations of
statics and dynamics. Balance equations. Conservation laws. Examples and
applications. Work, power, conservative forces. Examples. Mechanical
energy conservation theorem. Kinetic energy in motion around the center
of mass. König's theorem. Kinetic energy and angular momentum for a
rigid system. Examples and exercises.

**Elements of analytical mechanics**. Possible, virtual
and elementary displacements. Smooth constraints. Principle of reaction
forces. Examples. Symbolic equation of dynamic. Principle of virtual
work. Principle of stationary potential. Principle of Torricelli.
Lagrange equations. Integral of motion. Stability of equilibrium.
Theorems of Dirichlet and Liapunov (notes). Small oscillations around a
stable equilibrium position. Qualitative study of a conservative system
with one degree of freedom. Examples and exercises.

## Textbook Information

1. Carlo Cercignani, Spazio Tempo Movimento, Zanichelli

2. Lucio Demeio, Elementi di meccanica classica per l'ingegneria, Città Studi

3. G. Frosali, F. Ricci, Esercizi di Meccanica Razionale, Esculapio, Bologna

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Corpo rigido | Cercignani, Demeio |

2 | Calcolo vettoriale | Demeio |

3 | Baricentri e momenti d'inerzia | Demeio |

4 | Equazioni cardinali | Cercignani, Demeio |

5 | Equazioni di bilancio | Cercignani, Demeio |

6 | Statica del punto e dei sistemi | Cercignani, Demeio |

7 | Forze conservative | Cercignani, Demeio |

8 | Vincoli | Cercignani, Demeio |

9 | Principio dei lavori virtuali | Cercignani, Demeio |

10 | Condizioni di equilibrio | Cercignani, Demeio |

11 | Stabilità dell'equilibrio | Cercignani, Demeio |