# MATHEMATICAL PHYSICS II

Academic Year 2023/2024 - Teacher: Massimo TROVATO

## Expected Learning Outcomes

The course has as main objective the theoretical treatment of classical mechanics through the study of Analytical Mechanics. The course allows the student to connect the topics covered with the concepts learned in Calculus I, Calculus II, Geometry I, General Physics I and Mathematical Physics I.

The course of "Mathematical Physics II" aims to complete the program carried out in the course of "Mathematical Physics I".The course pursues as main objective is to treat classical mechanics, with more advanced mathematical tools such as modern methods variational.  Within  this approach, the equations of motion for a generic physical system will therefore be derived, starting from considerations geometric and symmetry properties of space-time, using general variational principles both in the space of the configurations that in the phase space. With this course will be analyze the deep connection between the geometric properties of a given physical system and the laws of physics that govern it. The variational principles have been, and are at present, the essential tools for the description of modern physics (classical and non-classical). To this end, as a final objective we will show how Analytical Mechanics can also be used to describe systems that are not necessarily mechanical (electromagnetic fields) and not necessarily classical (relativistic case). Thus, the final part of the course will be briefly dedicated to the study of variational principles applied, in event space, to the motion of a charged particle in an electromagnetic field, connecting this part of the program with the study of generalized potentials.

In particular, in reference also to the so-called "Dublin Descriptors", the course will aim to achieve the following transversal skills:

1) Knowledge and understanding:
One objective of the course shall be provide mathematic instruments, such as theorems and algoritms, which permit to face real problems in applied mathematics, physics, informatics, chemistry, economy and many other fields. With these mathematical instruments, student gets new abilities to clear useful theoretical and application problems.

2) Applying knowledge and understanding:
At the end of course student will be able to get new mathematical techniques of knowledge and understanding to face all possible links moreover, if it is possible, they will propose untreated new problems.

3) Making judgements:
Course is based on logical-deductive method which wants to give to students authonomus judgement useful to understanding incorrect method of demonstration also, by logical reasoning, student will be able to face not difficult problems, in applied mathematics, with teacher's help.

4) Communication skills:
In the final exam, student must show, for learned different mathematical techniques, an adapt maturity on oral communication.

5) Learning skills:
Students must acquire the skills necessary to undertake further studies (master's degree) with a high degree of autonomy. The course in addition to proposing theoretical arguments presents arguments  which also should be useful in different working fields.

## Course Structure

The lessons will be held through classroom. In these lessons the program will be divided into the following sections: "Short Connections with the Mathematical Physics I course"; “Variational principles in mechanics”; “Variational principles in the theory of electromagnetic fields”. In each of these sections first it will be discussed the main theoretical topics and then showed how these topics can be linked to possible applications. Then, many exercises are presented and discussed to identify solutions and applications on topics related to theoretical results.

The course is composed by 6 CFU of which:
5 CFU (corresponding to 7 hours each) are dedicated to theoretical lessons in the classroom for a total of 35 hours, e
1 CFU (corresponding to 12 hours) are dedicated to classroom exercises.
The course, of 6 CFU, therefore includes a total of 47 hours of teaching activities.

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 our Department, prof. Filippo Stanco

## Required Prerequisites

(Indispensable): Differential and integral calculus for functions of several real variables

(Indispensable): basic physical concepts of classical mechanics, in particular relating to kinematics and dynamics for systems of material points and for continuous 1D and 2D 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 Mathematics: in order to take the Mathematical Physics II exam it is necessary to have passed the exams of:  Analysis I, Analysis II, Physics I. Mathematical Physics I

## Attendance of Lessons

Attendance is strongly recommended

## Detailed Course Content

Brief summary and links with the course of Mathematical Physics I:
Vectorial and tensorial algebra. Pseudo-Euclidean and Euclidean spaces. Metric tensor. Covariant and contravariant components of a vector.  Coordinates Cartesian, polar, spherical and Cylindrical. Natural References. Metric induced by the transformation of coordinates. Curvilinear coordinates. Tensorial Algebra. Covariant, contravariant and mixed components of a tensor. Generalized potentials theory. Application to the case of an electromagnetic field and application to apparent forces. General notions on the configuration space. Tangent vectors and tangent space.

Variational principles Mechanics:
Variational principles and Lagrange equations in the Configuration Space. Tangent space. Hamilton Functional. First variation of Hamilton Functional. Gauge invariance of the variation of Hamilton functional and applications. Action. Maupertuis's principle of least action. The case of an isolated particle. Geodesic and correlation with the law of inertia.  The Brachistochrone problem. Connection between the Principle of the least action and Fermat's principle. Basics on De Broglie's theory. Phase space.  Dual space of tangent space. Hamiltonian formalism. Legendre transformations. Hamilton's equations. 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.  Connections between Canonical Transformations and the Gauge Transformations. Connection between Poisson's  brackets and Canonical Transformations. Hamilton-Jacobi theory. Connection between Hamilton-Jacobi theory and canonical transformations. Jacobi's theorem. Hamilton-Jacobi equation and its applications.  Two-body problem and explicit determination of the motion trajectories.

Variational principles in the theory of electromagnetic fields :

Introduction to the theory of special relativity. 4-dimensional formalism. Event as a point in spacetime. Non-Euclidean and Minkowski metrics. Types of 4-intervals. Lagrangian formulation and equations of motion deduced from variational principles.Variation of a functional in fields theory. Tensor of the Electromagnetic Field. Gauge invariance and its connection with potentials generalized. Invariants of the Electromagnetic Field. Construction of the Lagrangian function using the representation theorems for scalar functions of the Lorentz group. General formulation for the linear and nonlinear Maxwell equations, microscopic interpretation, experimental verification.

## Textbook Information

1. Teacher's notes.  (https://www.dmi.unict.it/trovato/PDF%20Fisica%20Matematica%20II%20AA%202020-2021.html)

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

## Course Planning

SubjectsText References
1Variational principles and Hamilton's principleTeacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
2Principle of least action. Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
3Symmetries and conservation laws, Noether's theorem.Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
4Problem of the two bodies.Teacher's notes, Goldstein, Landau-Lifsits Vol.1,
5Hamilton equations.Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
6Canonical transformationsTeacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
7Hamilton-Jacobi theory.Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti
8Poisson's bracketsTeacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein,
9Variational principles in electromagnetic field theoryTeacher's notes, Goldstein, Landau-Lifsits Vol.2

## Learning Assessment

### Learning Assessment Procedures

No ongoing tests will be carried out.

Verification of preparation is carried out only through oral exams, which take place during the periods provided in the academic calendars of the Department, on dates (exam sessions) published in the annual calendar of exam sessions (or Exam Calendar).

Verification of learning can also be carried out electronically, should the conditions require it.

In the criteria adopted for the evaluation of the oral exam, the following will be evaluated: