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Mechanics theory and non-linear systems

Code: 100172
Credits: 6
2026/2027
Degree programme Type Course
Physics OP 4

Contact lecturer

Name :
Santiago Peris Rodriguez
Email :
santiago.peris@uab.cat

Group languages

You can consult this information at the end of the document.

Prerequisites

 

It is advisable that the student has completed successfully a course on Classical Mechanics.

 

Re mathematical prerequisites, it is advisable that the student has previous knowledge of Calculus with a Complex Variable and Group Theory.

 

Objectives

The main goal in this course is to introduce the student to Theoretical Mechanics.

This introduction is supposed to give the student all the necessary knowledge which should be the basis for studying modern physics. 

In more concrete terms, these are the three main objectives:

  1.  To introduce the student to the different formalisms of Classical Mechanics: D'Alembert's formalism, Lagrange's, Hamilton's, canonical and Hamilton-Jacobi's; 
  2. To complete an adequate education of the student in the field of Classical Mechanics; 
  3. To introduce the student to Classical Field Theory.

Apart from the aforementioned goals, it will also be very important to estimulate a critical view in the student and to encourage a research-oriented attitude.

Learning outcomes

  1. Communicate complex information in an effective, clear and concise manner, either orally, in writing or through ICTs, in front of both specialist and general publics.
  2. Use critical reasoning, show analytical skills, correctly use technical language and develop logical arguments
  3. Work independently, take initiative itself, be able to organize to achieve results and to plan and execute a project.
  4. Working in groups, assume shared responsibilities and interact professionally and constructively with others, showing absolute respect for their rights.
  5. Describe the connection between dynamic equations and variational principles.
  6. Describe the relationship between symmetry and the law of conservation.
  7. Describe the concepts of displacement and virtual work.
  8. Describe properties of canonical transformations.
  9. Apply Lagrangian and Hamiltonian formalism to different physical systems to obtain equations of motion.
  10. Construct magnitudes conserved from Noether's theorem.
  11. Apply canonical transformations to obtain equations of motion.
  12. Apply ligation conditions within a system to find the relevant degrees of freedom and dynamic variables.
  13. Compare the applicability of the equations of motion and laws of conservation in different fields of science.
  14. Applying Lagrange and Hamilton formalism to discrete relativistic systems and to field theories describing the fundamental interactions of nature.
  15. Apply the method of canonical perturbation theory.
  16. Construct a Lagrangian based on the symmetries of the physical system.
  17. Use variational calculus.
  18. Use vector calculus and differential equations.

Contents


  1. D'Alembert's Formulation: Constraints. Virtual displacements. D'Alembert's principle. Generalized coordinates. Lagrange's equations.

  2. Lagrange's formulation: Calculus of variations. Hamilton's principle. Euler-Lagrange's equations. Extension to non-holonomic systems.

  3. Symmetries and conservation laws: Theorems of conservation: energy conservation, linear and angular momentum. Symmetry test. Noether's theorem. Symmetries in Classical Mechanics: Galileo's group.

  4. Hamilton's formulation: Phase space. Legendre's transformation.Hamilton's function. Canonical equations. Poisson's brackets.

  5. Hamilton-Jacobi's formulation: Method of separation of variables. Examples.

  6. Introduction to the Theory of Classical Fields: Lagrangian and Hamiltonian formulation of a continuous medium. Relativistic Field Theory. Examples. Symmetries and conservation laws in Classical Field Theory: energy-momentum tensor, Noether's theorem, internal and external symmetries. Examples.

Learning activities and methodology

Title Hours ECTS Learning outcomes
theory classes 33 1.32 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
Problem solving 47 1.88 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
Exercises calsses 16 0.64 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18
Study of the theory fundamentals 48 1.92 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

The work method will be divided between conducted teaching activities and self-teaching activities.

The conducted teaching activities will also be divided between theory classes in front of a blackboard together with tutorials where the students will be able to solve their difficulties, and exercise classes where the students will see how to apply all the concepts previously explained in class.

The self-teaching activities will consist in studying the theory foundations by the student and their application on the different examples, by means of solving exercises both at an individual level and at a group level.

Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
make-up exam 80% 3 0.12 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
mid-term exam 35% 1 0.04 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
Delivery of Exercises 20% 1 0.04 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
final exam 45% 1 0.04 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

Grading (Ordinary)

A) Take-home exercises (20% of the final grade): one or more exercises will be set, periodically, to be solved and handed in at a time that will be eventually established.

B) Mid-semester exam (35% of the final grade): it is a written exam, withour books, individual, about the middle of the semester.

C) Final exam (45% of the final grade): it is a written exam, without books, individual, at the end of the semester. The final grade will be the result of A+B+C.

D) Make-up exam of B+C: this exam is optional, without books, at the end of the semester. If the grade achieved from A+B+C > 3.5/10, the student will have the right to take this make-up final exam provided he/she has already

taken both exams B+C. The final grade achieved in this exam will replace the previous grade from B+C in all cases.

Grading (\"Avaluacio Unica\")

A) Final Exam (45% of the final grade): it is a written exam, without books, individual, at the end of the semester.

B)Examen Oral (55% of the final grade): it is an oral exam, individual, at the end of the semester.

C)Oral Make-up Exam (100% of the final grade): this exam is optional, at the end of the semester. If the grade achieved from A+B > 3.5/10, the student will have the right to take this make-up final exam provided he/she has already taken both exams A+B. The final grade achieved in this exam will replace the previous grade of A+B (Avaluacio Unica) in all cases.

Both evaluations (\"Unica\" and Ordinary) will have the final exams on the same day. Idem concerning the make-up exam.

The bonus points ("palotes") will not be added to the result of the make-up exam.

Bibliography

  1. Classical Mechanics, H. Goldstein, C. P. Poole i J. L. Safko, Addison Wesley (2002).
  2. Classical Mechanics: System of Particles and Hamiltonian Dynamics, W. Greiner, Springer-Verlag (2010).
  3. Classical Dynamics of Particles and Systems, J. B. Marion i S. T. Thornton, Brooks Cole (2004).
  4. Course in Theoretical Physics Vol. 1: Mechanics, L. D. Landau i E. M. Lifshitz, Butterworth-Heinemann (1995).
  5. Lectures in Analytical Mechanics, F. Gantmacher, Mir Publishers Moscow (1975).
  6. Mechanics: From Newton's Laws to Deterministic Chaos, F. Scheck, Springer-Verlag (2005).
  7. Mathematical Methods of Classical Mechanics, V. I. Arnold, Springer-Verlag (1989).
  8. An Introduction to Quantum Field Theory, M. E. Peskin i D. V. Schroeder, Perseus Books (1995).

Software

There isn't one.

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 English first semester morning-mixed
(PAUL) Classroom practices 1 English first semester morning-mixed