Analytical mechanics

2018/2019

Programme:

Physics, Second Cycle

Orientation:

Biophysics

Year:

1. in 2. year

Semester:

first

Kind:

optional

ECTS:

Language:

slovenian

Course director:

Prof. Dr. Tomaž Prosen, Assoc. Prof. Dr. Pavle Saksida

Hours per week – 1. semester:

Lectures

Seminar

Tutorial

Lab

Enrollment into the program.

Introduction to canonical formalism of classical (hamiltonian) mechanics.
Hamilton’s principle
Lagrange and Hamilton’s functions – Legendre transformation
Hamiltonian, Hamilton’s equations
Poisson brackets, Liouville equaiton
Generating functions and canonical transformations
Integrability, symmetries, Noether’s theorem, canonical action-angle variables, Hamilton-Jacobi equation
Examples: systems of coupled harmonic oscillators, tops
Hamiltonian systems with infinitely many degrees of freedon – Hamiltonian field theories
Symplectic manifolds and global description of hamiltonian systems

H. Goldstein, 'Classical Mechanics', 2nd ed., Addison Wesley, 1981,
L. N. Hand, J. D. Finch, Analytical Mechanics, Cambridge University Press, 1998,
V. I. Arnold,"Mathematical Methods of Classical Mechanics", (Springer-Verlag, New York 1978).

Canonical formulation of classical (hamiltonian) mechanics.

Knowledge and understanding:
Student should master the basics of classical mechanics based on the hamiltonain formalism.

Application:
Newly obtained knowledge should be used for a unified understanding of hamiltonian mechanics and mathematically correct formulation of solving practically relevant problems in mechanics, and also in related fields such as quantum mechanics, field theory and statistical physics.

Reflection:
Application of an abstract mathematical theory for a solid formulation of physics phenomena related to mechanics.

Transferable skills:
The methods and contents of the course represents the basis for a deeper understanding of classical mechanics, field theory and statistical physics.

Lectures, homework projects and their discussion jointly in the class

Completing a small homework project and its presentation in the class
Written exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

1) GORIN, Thomas, PROSEN, Tomaž, SELIGMAN, Thomas H., ŽNIDARIČ, Marko. Dynamics of Loschmidt echoes and fidelity decay. Physics reports, ISSN 0370-1573. [Print ed.], 2006, 435, nos. 2-5, str.3-156. [COBISS-SI-ID 1972068]
2) PROSEN, Tomaž. Open XXZ spin chain : nonequilibrium steady state and strict bound on ballistic transport. Physical review letters, ISSN 0031-9007. [Print ed.], 2011, vol. 106, issue 21, str. 217206-1-217206-4. [COBISS-SI-ID 2347108]
3) ILIEVSKI, Enej, PROSEN, Tomaž. Thermodynamic bounds on Drude weights in terms of almost-conserved quantities. Communications in Mathematical Physics, ISSN 0010-3616, 2013, vol. 318, no. 3, str. 809-830. [COBISS-SI-ID 2535524]
4) SAKSIDA, Pavle. On the nonlinear Fourier transform associated with periodic AKNS-ZS systems and its inverse. Journal of physics. A, Mathematical and theoretical, ISSN 1751-8113, 2013, vol. 46, no. 46, 465204 (22 str.). [COBISS-SI-ID 16833369]
5) SAKSIDA, Pavle. Lattices of Neumann oscillators and Maxwell-Bloch equations. Nonlinearity, ISSN 0951-7715, 2006, vol. 19, no. 3, str. 747-768. [COBISS-SI-ID 13932377]