# 12 - Theoretical Mechanics - Canonical Transformations I

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Javier Garcia
Published at : 14 Nov 2021
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Instructors: Santi Peris & Javier García
As Taught In: Fall 2020
Organization: Universitat Autònoma de Barcelona (UAB)

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Prerequisites:
It is advisable that the student has completed successfully a course on Classical Mechanics.
It is advisable that the student has previous knowledge of Calculus with a Complex Variable and Group Theory.

Objectives and Contextualisation
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, and canonical’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
- Apply Lagrangian and Hamiltonian formalism to different physical systems to obtain equations of motion.
- Apply canonical transformations to obtain equations of motion.
- Apply ligation conditions within a system to find the relevant degrees of freedom and dynamic variables.
- Apply the method of canonical perturbation theory.
- Applying Lagrange and Hamilton formalism to discrete relativistic systems and to field theories describing the fundamental interactions of nature.
- 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.
- Compare the applicability of the equations of motion and laws of conservation in different fields of science.
- Construct a Lagrangian based on the symmetries of the physical system.
- Construct magnitudes conserved from Noether’s theorem.
- Describe properties of canonical transformations.
- Describe the concepts of displacement and virtual work.
- Describe the connection between dynamic equations and variational principles.
- Describe the relationship between symmetry and the law of conservation.
- Use critical reasoning, show analytical skills, correctly use technical language and develop logical arguments
- Use variational calculus.
- Use vector calculus and differential equations.
- Work independently, take initiative itself, be able to organize to achieve results and to plan and execute a project.
- Working in groups, assume shared responsibilities and interact professionally and constructively with others, showing absolute respect for their rights.