# MECCANICA ANALITICAModule DIDATTICA FRONTALE

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.

With the "theoretical lessons" of Module "Didattica Frontale" the student will acquire 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 "think", by relating and linking the various topics covered and 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 "Didattica Frontale" of the Analytical Mechanics course will therefore aim to achieve the following transversal skills:

• Knowledge and understanding:

The Module "Didattica Frontale" aims to provide mathematical tools (such as theorems, demonstrative procedures and algorithms) that allow students to tackle real applications: in applied mathematics, physics, computer science and many other fields. With these tools, the student will acquire "new abilities to understand and describe" 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 "Didattica Frontale" 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 "Didattica Frontale", based on a logical-deductive method, will give the student autonomous judgment skills to discern incorrect methods of demonstrations. Furthermore, the student, through logical reasoning, will have to face adequate problems of mechanics, and more generally of applied mathematics, seeking to solve them with the interactive help of the teacher.

• Communication skills:
In the final exam, student must show that he has reached an adequate maturity in  oral communication, both the various mathematical techniques learned and 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, the course deals with topics that may be useful in various fields and professional settings.

## Course Structure

The section labeled as module "Didattica Frontale" of the Analytical Mechanics course will be carried out through theoretical frontal lessons carried out by the teacher in the classroom. In these "frontal theoretical lessons" the program will be divided into the sections reported in the "course content" for the module "Didattica Frontale" of Analytical Mechanics. In each of the frontal lessons the teacher will first address all the theoretical topics, showing how these topics can be connected to possible applications and specific physical problems.

The section of the Analytical Mechanics course consisting of the module "Didattica Frontale" is made up of 7 credits (corresponding to 7 hours each) for a sum up of 49 hours

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 (Theory, 4 hours):

Dimensions and bases of a vector space. Pseudo-Euclidean and Euclidean spaces. Metric tensor. Covariant and contravariant components of a vector. Tensor Algebra. Covariant, contravariant and mixed components of a tensor. Coordinate Changes. Tangent plane to a surface. Local mapping of a surface. Curvilinear coordinates. Differential operators in physics. Levi-Civita tensor.

Kinematics (Theory, 5 hours):

Particle kinematics. Curvilinear abscissa. Intrinsic reference systems. Frènet trihedron, torsion and curvature.  Frènet formulas. Velocity and acceleration of a point particle: plane motion, helical motion. Kinematics of rigid bodies. Poisson's formulas and angular velocity, degrees of freedom. Translational rigid motion. Rotational rigid motions. Plane rigid motions. Spherical rigid motions. Helical rigid motions. Relative kinematics. Composition of the velocities, of the accelerations and of the angular velocities.  Galilean equivalence. Drag and Coriolis accelerations. Theory of compound motions. Euler angles.

Dynamics and Lagrange equations for mechanical systems subjected to forces of any nature (Theory, 10 hours):

Axioms of classical dynamics. Cardinal equations and conservation laws. Rigid-body dynamics. Centers of mass and moments of inertia. Inertia tensor, principal axes, principal moments of inertia. Variation of the inertia tensor as the pole varies. Kinetic energy and angular momentum of a rigid body. Koenig's theorem. Holonomic and non-Holonomic constraints for physical systems. Pure rolling motion in the 1D and 2D cases. Generalized coordinates and degrees of freedom. Configuration space. Bilateral and unilateral constraints. Possible and virtual displacements. Principle of virtual work. Smooth constraints. Principle of d'Alembert. Lagrange equations. Conservative force fields and potentials. Conservation of energy. Generalized potentials. Integrals of motion. Equilibrium positions and their stability. Lyapunov theorem and Dirichlet theorem (statements) on stability.

Variational principles for mechanical systems subject to forces deriving from conservative potentials and/or generalized potentials (Theory, 30 hours):

Variational principles and Lagrange equations in the Configuration Space. Tangent space. Hamilton Functional. First variation of Hamilton Functional. Covariance of Lagrange equations under coordinates transformations. Gauge invariance of the variation of Hamilton functional and applications. Action. Maupertuis's principle of least action. Differences and similarities between Maupertuis's principle and Hamilton's Principle. The case of an isolated particle. Geodetic and correlation with the law of inertia. Explicit Calculations of the Geodetic. The Brachistochrone problem. Connection between the Principle of the least action and Fermat's principle. Basics on De Broglie's theory. Phase space. Conjugated moments and transformation laws. Dual space of tangent space. Hamiltonian formalism. Legendre transformations. Hamilton's equations. Derivation of Hamilton's equations from a variational principle. Application of Hamiltonian methods to various problems. Symmetries and conservation laws. Noether's theorem. Poisson brackets. Connection between Poisson brackets and conservation laws. Poisson's theorem. Canonical transformations. Cyclic variables. Canonical transformation induced by a pointwise transformation. Connection between canonical transformations and exact differential forms. Generating functions of a canonical transformation. Generating functions of type F1, F2, F3, F4. Determination of the more general generating function associated with a canonical transformation. Connections between Canonical Transformations and the Gauge Transformations. Connection between Poisson's  brackets and Canonical Transformations. Hamilton-Jacobi theory. Derivation of the Hamilton-Jacobi equation starting from a variational principle. Connection between Hamilton-Jacobi theory and canonical transformations. Jacobi's theorem. Hamilton-Jacobi equation and its applications. Separation of variables method for Hamilton-Jacobi equation. Two-body problem and explicit determination of the motion trajectories.

## Textbook Information

For the theoretical frontal lessons (7 CFU) the following will be used:

1. Teacher's notes.

2. S. Rionero, Lezioni di Meccanica razionale, Liguori Editore.
3. Strumia Alberto, Meccanica razionale. Vol. 1 e Vol. 2, Ed. Nautilus Bologna  (http://albertostrumia.it/?q=content/meccanica-razionale-parte-ii)
4. Strumia Alberto, Complementi di Meccanica Analitica                                     (http://albertostrumia.it/?q=content/meccanica-razionale-parte-ii)
5. A.Fasano, V.De Rienzo, A.Messina, Corso di Meccanica Razionale, Laterza, Bari.
6. H. Goldstein, Meccanica classica, Zanichelli, Bologna.
7. L.D. Landau E. M. Lifsits, Fisica teorica. Vol. 1: Meccanica, Editori Riuniti.
8. Valter Moretti, Elementi di Meccanica Razionale, Meccanica Analitica e Teoria della Stabilità. ( http://www.science.unitn.it/~moretti/runfismatI.pdf )

## Course Planning

SubjectsText References
1Composition of velocities and accelerationsTeacher's notes, Rionero, Strumia (Mecc. Raz.), Moretti
2Kinematics of rigid bodiesTeacher's notes, Rionero,Strumia (Mecc. Raz.), Moretti
3Euler's anglesTeacher's notes, Rionero,Strumia (Mecc. Raz.), Moretti
4Dynamics of rigid bodiesTeacher's notes, Rionero,Strumia (Mecc. Raz.), Moretti
5Lagrange equationsTeacher's notes, Rionero,Strumia (Mecc. Raz.), Moretti
6Generalized potentialsTeacher's notes, Rionero,Strumia (Mecc. Raz.), Moretti
7Variational principles and Hamilton's principleTeacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
8Principle of least actionTeacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
9Symmetries and conservation laws, Noether's theorem.Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
10Problem of the two bodies.Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
11Hamilton equations.Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
12Canonical transformationsTeacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
13Hamilton-Jacobi TheoryTeacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
14Poisson's bracketsTeacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti

## 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 relevance of the answers to the questions asked;
• 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, below for the exam, do not constitute an exhaustive list but represent  only a few examples:

Natural Reference Systems

Frenet formulas

Kinematics and dynamics of rigid motions

Euler's angles

Inertia tensor and its properties

Konig's theorem

Pure rolling motions in mechanics

Theory of holonomic, smooth and bilateral constraints

Lagrange equations

Generalized Potential Theory and applications

First integrals

Configuration space

Space tangent to a trajectory in the configuration space

Lagrangian systems

Trajectory deformations considering non-fixed extremes and non-synchronous trajectories

Hamilton functional and Hamilton principle

Gauge invariance of the first variation of Hamilton functional, applications

Functional Action, Isoenergetic Deformations and Maupertuis Principle of Least Action.

Geodesics and applications

Brachistocrona theory

Connection between the Principle of Least Action and Fermat's Principle

Symmetries and conservation laws, Noether's theorem.

Phase space, Hamilton equations and applications.

Canonical transformations and application examples

Hamilton-Jacobi theory and application examples

Poisson brackets and connection with the computation of prime integrals

Poisson's parenthesis and connection with canonical transformations

Two-body problem

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