# MATHEMATICAL PHYSICS I

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

**Orazio MUSCATO**

## Expected Learning Outcomes

To provide basic knowledge of vector calculus, statics and dynamics of material systems and rigid bodies.

## 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 di 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 |

## Learning Assessment

### Learning Assessment Procedures

The exam takes place through a written and an oral test. The written test is preparatory to the oral one. The (final) oral test will verify the theoretical and practical knowledge of the topics covered during the course. The evaluation of the exam is based on the following criteria: level of knowledge of the required topics, expressive ability and teaching topics, ability to apply knowledge to simple case studies, ability to connect the different themes of the teaching program. During the year, eight exam sessions are scheduled as per the academic calendar, in which both the written and oral tests can be taken. In the periods allowed by the academic calendar, beyond the reception hours, it is possible, by contacting the teacher via e-mail, to arrange further meetings with the teacher. Verification of learning can also be carried out electronically should the conditions require it, request it.