# MECCANICA ANALITICA 1Module ESERCITAZIONI

Academic Year 2023/2024 - Teacher: Massimo TROVATO

## Expected Learning Outcomes

The course has as its main objective the theoretical treatment of classical mechanics using, appropriately, differential calculus, integral calculus and variational calculus with basic elements of differential geometry. The course program is divided into sections:

"Vector and tensor algebra", "Kinematics", "Dynamics and Lagrange equations for mechanical systems subjected to forces of any nature", "Variational principles for mechanical systems subject to forces deriving from conservative potentials and/or generalized potentials".

All sections are closely interconnected and necessary for the purpose of understanding the entire course of Analytical Mechanics. Specifically, the course consists of theoretical lessons, but presents also numerous exercises and applications.

NoteThe lessons of the Analytical Mechanics course (9 CFU) are formally contained both in the module "Didattica Frontale" (7 CFU) and in the module "Exercises" (2 CFU). This separation is only formal since, during the lessons held in the classroom, the teacher will first address the theoretical topics of module "Didattica Frontale", integrating these topics with the contents provided for the module "Exercises". Obviously, the number of hours required for the corresponding number of CFU is dedicated to the topics contained in the two modules.

With the lessons of module "Exercises" the student will acquire and apply the basic "theoretical knowledge" for:

i) the study of mechanical systems with particular attention to the kinematics and dynamics of rigid material systems subjected to forces of any nature.

ii) the study of variational methods aimed at describing the physics of material systems subject to forces deriving from conservative potentials and/or generalized potentials.

iii) The study of conservation laws in physics and their connection with the symmetry properties of the physical system considered.

iv) the description of the laws of physics, when possible, in geometric terms.

v) the possibility of "solving" (also with "successive approximation" methods) the equations of motion, explicitly determining the evolutionary solutions for the physical system considered.

The objective of the course is to induce the student to apply, "with exercises" the concepts of Analytical mechanics, in order to relate and connect the various topics covered to each other, acquiring new knowledge and skills.

To this end, according to with the teaching regulations of the CdS in Physics, it is expected that at the end of the course the student will have acquired:

- inductive and deductive reasoning ability;

- ability to schematize natural phenomena in terms of physical quantities, to set up a problem using suitable relationships (of an algebraic, integral or differential type) between physical quantities and to solve the problems with analytical and/or numerical methods;

- ability to understand simple experimental configurations in order to carry out measurements and analyze data.

The course will allow the student to acquire useful skills for various technical-professional opportunities, and in particular:

- For technological applications in the industrial and training sectors.

- For the acquisition and processing of data.

- To take care of modeling activities, analysis and related IT-physical implications.

Specifically, having to express the "expected learning outcomes", through the so-called "Dublin Descriptors", the Module of "Exercises" of the Analytical Mechanics course will therefore aim to achieve the following transversal skills:

• Knowledge and understanding:

The module of "Exercises" aims to use and apply mathematical tools (such as theorems, demonstrative procedures and algorithms) to describe real problems in classical mechanics. The student with these tools will have "new skills to understand, describe and solve" the mathematical schemes hidden behind the physical processes studied during the course. This knowledge will also be very useful to understand new theoretical problems, which can be addressed both in subsequent studies and in the real world.

• Ability to apply knowledge and understanding:

At the end of Module of "Exercises" the student will be to acquire the "ability to apply knowledge and understanding" of the new mathematical techniques studied, both to concretely determine the "solutions of the equations of motion" associated with the physical problems studied during the course, and for the concrete resolution of possible new problems untreated during the course.

• Independent judgment:

The Module of "Exercises" of course, will give the student autonomous judgment skills to discern incorrect methods of solving the exercises. Furthermore, the student, through logical reasoning, will have to face adequate problems of mechanics, and more generally of applied mathematics.

• Communication skills:

In the final exam the student must show that he has reached an adequate maturity in  oral and written communication, both to describe the various mathematical techniques learned and to solve the physical problems described during the course.

• Learning ability:

Students will be able to acquire the skills necessary to undertake subsequent studies (master's degree) with a high degree of autonomy. In addition to proposing theoretical topics for solving the exercises, the course deals with topics that may be useful in various fields and professional settings.

## Course Structure

The part labeled as, Module of "Exercises" of the Analytical Mechanics teaching will be carried out through lessons held by the teacher in the classroom. For the "lessons" of the exercises, the program carried out will be completely integrated with the topics of the theoretical lessons of Module "Didattica Frontale" of the Analytical Mechanics course. In each of the lessons held in the classroom, the teacher will first address all the theoretical topics of Module "Didattica Frontale", integrating these lessons with suitable exercises. In this way the teacher will concretely show how the exercises are connected with the theory, describing both possible applications and specific physical problems.

Should the circumstances require online or blended teaching, appropriate modifications to what is hereby stated may be introduced, in order to achieve the main objectives of the course.

Exams may take place online, depending on circumstances.

Information for students with disabilities and/or DSA.

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs.

It is also possible to contact the referent teacher CINAP (Center for Active and Participated Integration - Services for Disabilities and / or DSA) of the Physics Department

## Required Prerequisites

The student must know:

(Indispensable): Trigonometry and elementary trigonometric relations, resolution of elementary algebraic and trigonometric equations, resolution of algebraic and trigonometric inequalities, vectors, matrices, resolution of linear systems, resolution of systems of inequalities, graphic representation in the plane of 1D functions and their inverse functions, elementary geometry in the plane, in space and their applications to elementary geometric objects (Straight lines, triangles, circles, ellipses, spheres, cylinders, cones, etc ...).

(Indispensable): differential and integral calculus for 1D functions.

(Indispensable): basic physical concepts of classical mechanics, in particular relating to kinematics and dynamics for systems of material points and for continuous 1D systems.

(Important): symbology associated with physical quantities, dimensions, and systems of units of measurement.

(useful): nomenclature and language properties for the elementary physical description of classical mechanics

As required by the teaching regulations of CdS in Physics: in order to take the Analytical Mechanics exam it is necessary to have passed the exams of: Mathematical Analysis I, Physics I.

## Attendance of Lessons

Attendance to the course is usually  mandatory (consult the Academic Regulations of the Course of Studies)

Attendance signatures may be collected during the lessons.

## Detailed Course Content

Vector and tensor algebra (Exercises, 4 hours):

Exercises on basis changes for vectors and double tensors. Exercises on the external product of vector spaces. Metric matrix induced by a change of coordinates from Cartesian to: polar, cylindrical and spherical. Exercises on: vector products, mixed product, double vector product, Levi-Civita tensor.

Kinematics (Exercises, 3 hours):

Calculation of the torsion and curvature of a cylindrical helix. Kinematics from a Cartesian coordinate system to Natural coordinate system. Explicit calculation of position, velocity and acceleration in the case of: rotational rigid motion and helical rigid motion. Calculation of the drag acceleration in the case of a rotational motion. Calculation of the angular velocity of a rigid body in terms of Euler angles.

Dynamics and Lagrange equations for mechanical systems subjected to forces of any nature (Exercises, 12 hours):

Exercises on: Centers of mass, inertia moments and Inertia tensors. Calculation of Kinetic Energy for a physical pendulum in R3 using the inertia tensor. Explicit calculation of potentials associated with conservative forces. Exercises on the calculation of generalized potentials. Explicit resolution of written exams tasks.

Variational principles for mechanical systems subject to forces deriving from conservative potentials and/or generalized potentials (Exercises, 11 hours):

Calculation of variations to determine the geodesic: in the plane, on a cylindrical surface, on a spherical surface. Exercises on the calculation of the Hamiltonian using the Legendre transform. Exercises on conservation laws using Poisson brackets. Exercises on canonical transformations using generating functions. Exercises on canonical transformations induced by a Gauge transformation. Exercises on solving the Hamilton-Jacobi equation: 1D harmonic oscillator, particle subjected to a central potential, particle in R2 subjected to a dipole field.

## Textbook Information

1.Appunti del docente.

2. S. Rionero, Lezioni di Meccanica razionale, Liguori Editore.

3. S. RioneroG. GaldiM. MaiellaroEsercizi e complementi di meccanica razionale (Vol. 2) Liguori Editore.

4.G. Borgioli, Dispensa di Meccanica Razionale https://www.modmat.unifi.it/upload/sub/Borgioli/Meccanica%20Razionale/dispense_mecraz.pdf

5. S. Siboni, Dispense per i corsi di Meccanica Razionale 1 e 2.

6. H. Goldstein, Meccanica classica, Zanichelli, Bologna.

7. L.D. Landau E. M. Lifsits, Fisica teorica. Vol. 1: Meccanica, Editori Riuniti

## Course Planning

SubjectsText References
1Vector and tensor algebraTeacher's notes, Texts 2-4
2KinematicsTeacher's notes, Texts 2-4
3Dynamics and Lagrange equations for mechanical systems subjected to forces of any natureTeacher's notes, Texts 2-3,5
4Variational principles for mechanical systems subject to forces deriving from conservative potentials and/or generalized potentialsTeacher's notes, Texts 6,7

## Learning Assessment

### Learning Assessment Procedures

No intermediate tests will be carried out before the end of course.

Verification of preparation is carried out through written exams and oral exams, which take place separately during the periods provided in the academic calendars of the Department, on dates (exam sessions) published in the annual calendar of exam sessions (https://www.dfa.unict.it/corsi/L-30/esami). In particular, the result of the written exam will contribute to the determination of the final grade after the completion of the oral exam.

Verification of learning can also be carried out via an online telematic connection, should the conditions require it.

For written exams:

1) The duration of each written exam is 3 hours;

2) It consists of a classical mechanics problem consisting of 3-4 exercises;

3) The contents of the chapters labeled as "Kinematics" and "Dynamics and Lagrange equations for mechanical systems subjected to forces of any nature" reported in the "Course contents" can be topics of the written test;

4) During the written exam students will only be able to use a trigonometric formulary and calculators, but not mobile phones

5) The written test has a duration of validity of two sessions (the one relating to the test carried out and the subsequent one), of the relative Academic Year. Therefore the exam session dates refer to the written test.

Results of  written exams:

The written exam is preparatory for the oral exam. To access the oral exam the student must solve at least two of the assigned exercises. Specific scores will not be given, but the three grades of judgment will be given

• Passed
• Passed with reserve
• Passed with a lot of reserve

Evaluation criteria for written exams:

• The student's operational ability to use mathematical tools for solving the exercises will be evaluated.
• The student's ability to apply theoretical knowledge of classical mechanics for solving the exercises will be evaluated.

For the ORAL exams:

1. The teacher may ask for clarifications or remarks about the written tests;
2. In the criteria adopted for the evaluation of the oral exam, the following will be considered:

• the level of detail of the contents displayed;
• the ability to connect with other topics covered by the program and with previously acquired topics, the ability to report examples;
• Correct use of technical language and clarity of presentation.

### Examples of frequently asked questions and / or exercises

The questions formulated in the oral exam, on exercises carried out during the course, do not constitute an exhaustive list but represent only some examples:

Transformation of coordinates from Cartesian to: cylindrical and spherical, transformation matrix and connection with the metric matrix. Natural reference frame systems. Determination of the tangent plane to a spherical surface and to a cylindrical surface.

Calculation of the torsion and curvature of a cylindrical helix.

Kinematics from a Cartesian reference to Natural reference frame systems.

Calculation of position, velocity and acceleration in the case of: rigid rotary motion and rigid helical motion.

Calculation of the angular velocity of a rigid body in terms of Euler angles.

Exercises on: Centers of mass, inertia moments and Inertia tensors.

Calculation of Kinetic Energy for a physical pendulum in R3 using the inertia tensor.

Generalized potential of apparent forces and applications.

Generalized potential of the electromagnetic field and applications to the motion of a charged particle.

Determination of the geodesic in the plane, on a cylindrical surface, on a spherical surface.

Given the Lagrangian, calculate the Hamiltonian using the Legendre transform for the following problems: Charged particle in an electromagnetic field. System of N rigid bodies subject to conservative and generalized potentials. Particle bound to a sphere subjected to a spherically symmetric potential.

Exercises and examples on canonical transformations. Determination of the Generating Functions associated with the following transformations: Punctual transformation. Identical transformation. Exchange transformation. Gauge transformation. 1D Harmonic Oscillator Problem.

Exercises on solving the Hamilton-Jacobi equation:: 1D harmonic oscillator. Particle in a central field in R2 and R3. Particle in R2 subjected to a dipole field.

The exercises, for the exam, shown in the link below, do not constitute an exhaustive list but represent only a few examples

https://www.dmi.unict.it/trovato/Testi_compiti.pdf