Статус: Дисциплина общефакультетского пула
Кто читает: Факультет математики
Где читается: Факультет математики
Когда читается: 1, 2 модуль
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.
- 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.
- 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
- Lagrangian formalismLagrangian formalism: Least action principle; Euler – Lagrange equations; first integrals and symmetries of action.
- Basics of Hamiltonian formalismBasics 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 integrabilitySeparations of variables and integrability: Hamitonian – Jacobi equations; canonical transformations. Moment map. Arnold -– Liouville integrable systems. Lax representation.
- Промежуточная аттестация (2 модуль)0.7 * final exam + 0.3 * problem sheets + midterms
- 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
- Jovanovic, B. (2016). Noether symmetries and integrability in time-dependent Hamiltonian mechanics. https://doi.org/10.2298/TAM160121009J