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Обычная версия сайта
2019/2020

Гамильтонова механика

Статус: Дисциплина общефакультетского пула
Когда читается: 1, 2 модуль
Язык: английский
Кредиты: 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

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

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

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

Assessment Elements

  • non-blocking problem sheets + midterms
  • non-blocking final exam
Interim Assessment

Interim Assessment

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

Bibliography

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

Recommended Additional Bibliography

  • Jovanovic, B. (2016). Noether symmetries and integrability in time-dependent Hamiltonian mechanics. https://doi.org/10.2298/TAM160121009J