Materials to the supplemantary seminars on "Advanced mechanis"

02.02.2009
Unfortunately this is the last seminar in this semester. On this seminar Alexei moved forward with details of Hamilton formalism for strings. Also some basic solution of equations of motion are considered. Notes written by Alexei are here. If somebody is interested in continuation of this seminar, please contact Maxim.

26.01.2009
Alexei started with classical string theory. He introduced the notion of Lorentz invariance and on basis of this invariance he demostrated how one constructs the Lagrange function for relativistic particle. Natural generalization of this Lagrange function is the that for a string - one dimensional object. He also started to discuss the general symmetries of Lagrange function for the string, in particular it was shown that the ends of the string move always with speed of light. Notes written by Alexei are here. Nowadays quantum string theory is considered as beeing a good candidate for Theory of Everything. Look popular videos about history of string theory on youtube - string-1, string-2, string-3, etc. (in total several videos).

19.01.2009
On this seminar Alena explained how to compute the regions of parametric resonances. Notes written by Alena are here.

12.01.2009
We continued with parametric resonance. Important notion of mapping for a period (monodromy matrix) was discussed. The conditions for a parametric resonance were derived. To do that the notion of Hamilton flows and Liouville theorem were important! Notes written by Alena are here.

05.01.2009
Christmas holidays is over and we continue with the conservation laws for the KdV equation - here we came to the existence of INFINITE number of conservation laws. This is an example that the conservation laws are not related to the symmetries of action! The derivation of these extra conservation laws requires the knowledge of quantum mechanics - hopefully we shall be able to discuss these elegant methods in future seminars on Advanced Quantum Mechanics! Also, we have started a new topic - parametric resonances, the phenomenon we meet very frequently in everyday life (just recall swinging)! Notes written by Alena are here.

15.12.2008
The two-soliton solution is presented and studied. The Lagrange function for the KdV equation is derived. Notes written by Alena are here.

08.12.2008
The one-soliton solution of the KdV equation is obtained! Notes written by Alena are here. In order to get the one-soliton solution we have reduced the KdV equation to the effective mechanical problem with one degree of freedom. Next Monday we shall play with the soliton solutions! We shall see that the soliton behaves as a particle. This observation has deep consequences - one can hypothesize that elementary particles (like protons and neutrons) are solitons. Such picture of elementary particles allowed us to predict a new exotic particle Theta+. Today came the news that the existence of this particle is confirmed! See news on our site!

01.12.2008
The problem from the previous seminar is solved with the help of the functional technique, also we reduced the Fermi-Pasta-Ulam chain to the famous KdV equation. Notes written by Alexei are here.

24.11.2008
We have started with solitons! Here is the Power Point presentation about a brief history of the solitons. We started with working out the mathematical formalism for the solving of Fermi-Pasta-Ulam chain. Script written by Alena is here. Problems to the first seminar can be downloaded here.