# Analytical mechanics

**Academic Year 2022/2023**- Teacher:

**Massimo TROVATO**

## Expected Learning Outcomes

**The course has as main objective the theoretical treatment of classical mechanics allowing the student to connect the topics of the course with the concepts learned in Calculus I, Calculus II, Geometry I, General Physics I and General Physics II.**

__With this course the student will acquire the basic knowledge for__:

i) The study of holonomic systems with particular regard to the kinematics and dynamics of rigid bodies.

ii) The Analytical Mechanics.

**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 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: Vectorial and tensorial algebra; Kinematics; Dynamics; Analytical 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 9 CFU of which:

7 CFU (corresponding to 7 hours each) are dedicated to theoretical lessons in the classroom for a total of 49 hours, and

2 CFU (corresponding to 15 hours each) are dedicated to classroom exercises, for a total of 30 hours.

The course (9 CFU) therefore includes a total of about 80 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 the Physics Department

## Required Prerequisites

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

**Vectorial and tensorial algebra:**

Vector spaces, dimensions and bases of a vector space. Pseudo-Euclidean and Euclidean spaces. Metric tensor, covariant and contravariant components. Coordinates Cartesian, polar, spherical and Cylindrical. Coordinate changes. Curvilinear coordinates. calculus scalar and vector products, mixed products. Tensorial Algebra. Covariant, contravariant and mixed components of a tensor. Vector fields in physics

**Kinematics:**

Particle kinematics. Curvilinear abscissa. Intrinsic systems of references. Osculator plane, osculator circle, torsion and curvature. FrÃ©net formulas. Motion, velocity and acceleration of a point particle: plane, circular, harmonic and helical motions. Kinematics of rigid bodies. Poisson's formulas and angular velocity. Analysis of the field of velocity of a rigid body. Different kinds of rigid motions. Plane rigid motions. Rigid body with a fixed point. Rigid body with a fixed axis. Rigid helical motion. Mechanics of rigid bodies, some applications. Relative kinematics. Composition of the velocities, of the accelerations and of the angular velocities. Galileian equivalence. Inertial frames and Galilei transformations. Inertial and not inertial systems of references. Coriolis theorem. Fictitious forces. Coriolis forces. Euler angles.

**Dynamics:**

Axioms of classical dynamics. Statics and the dynamics of a particle. Statics and the dynamics of a system. Cardinal equations in static and in dynamics. Conservation theorems. Rigid-body dynamics. Centers of mass and moments of inertia. Inertia tensor, principal axes. Principal moments of inertia. Properties of the inertia tensor. Huygens and Steiner's theorems. Koenig's theorem for the kinetic energy. Kinetic energy and angular momentum of a rigid body. Potential energy. Constraints. Holonomic and non-Holonomic constraints for physical systems. Generalized coordinates and degrees of freedom. Configuration space. Bilateral and unilateral constraints. Reversible and irreversible displacements. Ideal constraints. Possible and virtual displacements. Principle of virtual work. Principle of d'Alembert. Lagrangian and Lagrange equations. Conservative force fields and potentials. Conservation of energy. Generalized potentials and applications. Integrals of motion. Equilibrium positions. Stability of equilibrium positions. Lyapunov theorem. Dirichlet Stability Theorem. Linearized motions. Small oscillations around stable equilibrium points.

**Analytical Mechanics**:

Variational principles and the Lagrange equations. Configuration space. Tangent vectors and tangent space. Variational principle and Hamilton principle in the Configuration space. Principle of the least action and the Lagrange equations. Cyclic variables. Geodetic calculations. The brachistochrone problem. Connection between the Principle of least action and Fermat's principle. Wave description of a particle, notes on De Broglie's theory. Conserved quantities and Noether theorem. Two-body problem. Phase space. Hamiltonian Formalism. Legendre transformations. Hamilton equations. Derivation of Hamilton equations from a Variational principle. Application of Hamiltonian methods to various problems. Canonical transformations. Generating function of a canonical transformation. Applications and examples. The theory of Hamilton-Jacobi. Derivation of Hamilton-Jacobi equation from a Variational principle. Equation of Hamilton-Jacobi and its application. Variable separation in the method of Hamilton-Jacobi. Poisson brackets. Connection between Poisson brackets and conservation laws. Poisson theorem. Applications and examples. Connection between Poisson's brackets and Canonical Transformations.

**Variational principles in the theory of electromagnetic fields**:

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.

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 )

9. L.D. Landau E. M. Lifsits, Fisica teorica. Vol. 2: Teoria dei campi, Editori Riuniti.

## Course Planning

Subjects | Text References | |

1 | Composition of velocities and accelerations | Teacher's notes, Rionero, Strumia (Mecc. Raz.), Moretti |

2 | Relative kinematics | Teacher's notes, Rionero, Strumia (Mecc. Raz.), Moretti |

3 | Kinematics of rigid bodies | Teacher's notes, Rionero,Strumia (Mecc. Raz.), Moretti |

4 | Euler's angles | Teacher's notes, Rionero,Strumia (Mecc. Raz.), Moretti |

5 | Cardinal equations | Teacher's notes, Rionero,Strumia (Mecc. Raz.), Moretti |

6 | Dynamics of rigid bodies | Teacher's notes, Rionero,Strumia (Mecc. Raz.), Moretti |

7 | Lagrange equations | Teacher's notes, Rionero,Strumia (Mecc. Raz.), Moretti |

8 | Generalized potentials | Teacher's notes, Rionero,Strumia (Mecc. Raz.), Moretti |

9 | Variational principles and Hamilton's principle | Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti |

10 | Principle of least action | Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti |

11 | Symmetries and conservation laws, Noether's theorem. | Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti |

12 | Problem of the two bodies. | Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti |

13 | Hamilton equations. | Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti |

14 | Canonical transformations | Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti |

15 | Hamilton-Jacobi Theory | Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti |

16 | Poisson's brackets | Teacher's notes, Strumia (Compl. Mecc. Anal.), Goldstein, Landau-Lifsits Vol.1,Moretti |

17 | Variational principles in electromagnetic field theory | Teacher's notes, Landau-Lifsits Vol.2 |

## Learning Assessment Procedures

No ongoing tests will be carried out.

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 (or Exam Calendar). 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" reported in the "Course contents" can be topics of the written test;

4) 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.

__Results of written exams__:

The written exam is preparatory for the oral exam, specific scores will not be given, but the three grades of judgment will be given

- Approved
- Approved with reserve
- Approved 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:

- Possible comments relating to the written test are foreseen;
- In the criteria adopted for the evaluation of the oral exam, the following will be evaluated:

- 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 topics already acquired in previous years courses, the ability to report examples,
- the property of language and clarity of presentation.

## Examples of frequently asked questions and / or exercises

The exercises, for the written 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

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

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