Mathematical Methods of Physics

To undertand the Lagrangian and Hamiltonian formulation of Classical Mechanics.

To use Calculus on manifolds to give a desripcion of these formulations. This allow to obtain the solutions of the equations of Mechanics as integral curves of certain vector fields ssociated to the hamiltonian function (lagrangian).

To know the fundaments of Symplectic Geometry related to Classical Mechanics.

To study the relation between symmetries of equations of Mechanics and the motions constants.

1. Classical Mechanics: Lagrangian formulation on euclidean spaces. (2 hours)

2. Lagrangian formulation for holonomic systems. (3 hours)

3. Lagrangian Mechanics on manifolds. (4 hours)

4. Classical Mechanics: Hamltonian formulation on euclidean spaces. (2 hours)

5. Hamiltonian formulation for holonomic systems. (3 hours)

6. Hamiltonian Mechanics on manifolds. (4 hours)

7. Symmetries and motions constants. (2 hours)

8. Action of Lie groups on manifolds: momentum map. (4 hours)

Basic
R. Abraham e J.E. Marsden, Foundations of Mechanics, Benjamin, New York, 1978.

Complementary:
J. E. Marsden e T.S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, New York 1994.
W. D. Curtis e F. R. Miller. Differential manifolds and Theorical Physics
Manuel de León; Paulo R. Rodrigues
Methods of differential geometry in analytical mechanics, North-Holland Math. Studies, 158, 1989.
V. Arnold, Mathematical Methods of Classical Mechanics, GTM 60, Spriner-Verlag 1984

• 3.1 BASIC AND GENERAL COMPETENCES
• GENERAL
• • CG01 - Introduce students into the research, as an integral part of a deep formation, preparing them for the eventual completion of a doctoral thesis.
• • CG02 - Acquisition of high level mathematical tools for diverse applications covering the expectations of graduates in mathematics and other basic sciences.
• • CG03 - Know the broad panorama of current mathematics, both in its lines of research, as well as in methodologies, resources and problems it addresses in various fields
• • CG04 - Train for the analysis, formulation and resolution of problems in new or unfamiliar environments, within broader contexts.
• • CG05 - Prepare for decision making based on abstract considerations, to organize and plan and to solve complex issues.
• BASICS
• • CB6 - Possess and understand knowledge that provides a basis or opportunity to be original in the development and / or application of ideas, often in a research context
• • CB7 - That students know how to apply the knowledge acquired and their ability to solve problems in new or unfamiliar environments within broader (or multidisciplinary) contexts related to their area of study.
• • CB8 - That students are able to integrate knowledge and face the complexity of making judgments based on information that, being incomplete or limited, includes reflections on social and ethical responsibilities linked to the application of their knowledge and judgments
• • CB9 - That students know how to communicate their conclusions and the knowledge and ultimate reasons that sustain them to specialized and non-specialized audiences in a clear and unambiguous way
• • CB10 - That students have the learning skills that allow them to continue studying in a way that will be largely self-directed or autonomous.
• 3.2 TRANSVERSAL COMPETENCES
• • CT01 - Use bibliography and search tools for general and specific bibliographic resources of Mathematics, including Internet access
• • CT02 - Optimally manage work time and organize available resources, establishing priorities, alternative paths and identifying logical errors in decision making
• • CT03 - Enhance capacity for work in cooperative and multidisciplinary environments.
• 3.3 SPECIFIC COMPETENCES
• • CE01 - Train for the study and research in mathematical theories in development.
• • CE02 - Apply the tools of mathematics in various fields of science, technology and social sciences
• • CE03 - Develop the necessary skills for the transmission of mathematics, oral and written, both in regard to formal correction, as well as in terms of communicative effectiveness, emphasizing the use of appropriate ICT

From concrete physical examples we develop the geometric formulation of Classical Mechanics

Continuous evaluation through expositions of topics related to the subject and final exam, which will be fundamentally theoretical.


Continuous assessment note: 25% presentations, 75% final exam.

The student's grade will be the maximum between the continuous evaluation grade and the final exam grade

22 hours of lectures

2 hours os study in small groups

28 hours of personal study

5 hours to write exercices or other works.

3 hourssof computer work