• A
• A
• A
• ABC
• ABC
• ABC
• А
• А
• А
• А
• А
Regular version of the site
2019/2020

## Hamiltonian Mechanics

Type: Optional course (faculty)
When: 1, 2 module
Language: English
ECTS credits: 6

### Course Syllabus

#### Abstract

This is one of the basic theoretical physics courses for students in their 3-–4 year of undergraduate studies and for Masters students. A core of mathematical methods of modern theory of Hamiltonian systems are concepts created in various branches of mathematics: the theory of differential equations and dynamical systems; the theory of Lie groups and Lie algebras and their representations; the theory of smooth maps of manifolds. Many modern mathematical theories, such as symplectic geometry and theory of integrable systems have arisen from problems of classical mechanics. That’s why this course is recommended not only for those who plan to continue their studies in «Mathematical Physics» master program, but also for those who are planing to continue pure mathematical studies. #### Learning Objectives

• To gain understanding of basic concepts of modern theory of Hamitonian mechanics and symplectic geometry and skills in solving particular problems central for various applications. Additional attention will be paid to the theory of integrable systems. #### Expected Learning Outcomes

• To know how to solve equations of motion of basic system, to understand a concept of symplectic geometry, canonical transformations
• To appreciate a concept of Lax representation as a foundation of the theory of integrable systems #### Course Contents

• Lagrangian formalism
Lagrangian formalism: Least action principle; Euler – Lagrange equations; first integrals and symmetries of action.
• Basics of Hamiltonian formalism
Basics of Hamiltonian formalism: phase space; Legendre transform; Poisson brackets and symplectic structure; Darboux theorem, Hamiltonian equations.
• Examples: Geodesics on Lie groups. Mechanics of solid body and hydrodynamics of ideal fluid.
• Separations of variables and integrability
Separations of variables and integrability: Hamitonian – Jacobi equations; canonical transformations. Moment map. Arnold -– Liouville integrable systems. Lax representation. #### Assessment Elements

• problem sheets + midterms
• final exam #### Interim Assessment

• Interim assessment (2 module)
0.7 * final exam + 0.3 * problem sheets + midterms #### Recommended Core Bibliography

• Cortés, V., & Haupt, A. S. (2016). Lecture Notes on Mathematical Methods of Classical Physics. https://doi.org/10.1007/978-3-319-56463-0
• Landau, L. D., & Lifshitz, E. M. (2013). Course of Theoretical Physics (Vol. 3d edition). Saint Louis: Pergamon. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1443758