Loggbok SI1142 VT10
Advise on what to read:
L1-L4: [Var]; [BF] Chapter 2; [Tong] 2.1-2.3, 4.1
L5-L6: [Gr]; [BF] Chapter 7-8; [KS] Chapter 5
L7-L12: [Tong] Chapters 2 and 4; I also recommend that you get and study either [LL] (concise) or [G] (longer text).
Lektion 1: Introduction and overview: what is the course about (see e.g. the WIKIPEDIA pages on variational calculus and Green's functions). Variational problems (examples). Hamilton's principle. Euler-Lagrange equations ([Var] p1-6 and/ or [KS] Section 8.1). Derivation of Euler-Lagrange equations.
Lektion 2: How to solve Euler-Lagrange equations in practice. Remarks on boundary conditions. Variational problems with constraints (Lagrange multiplier method). Functionals depending on several variables. Variational problems for functions in several variables. Variational calculus for functions in several variables.
Lektion 3: Derivation of equations of motion for vibrating string using Hamilton's principle; same for membran (see e.g. [KS] Section 1.5). Introduction to analytical mechanics: Lagrange's formalism and examples. Hamiltonian formalism (outline). Comments: I my lectures I presented several examples from analytical mechanics. You can find an overview on Wikipedia. A detailed description, including the examples I discussed and a lot more, can be found in the lecture notes on David Tong (I'll refer to is as [T]) which I highly recommend. In these notes you can also find a list of standard textbooks on classical mechanics.
In the 2nd part of this course we will go into analytical mechanics in more detail.
Lektion 4: The structure of a physical theory: classical vs. quantum mechanics. Observables; states; time evolution; Poisson brackets. Comment: Much of what I discussed last time and today can be found in [T], 4.1-4.3. A
Lektion 5: Green's function for the Poisson eq. in 3D: basic idea ([KS] 5.1); fundamental solutions in 2D and 3D ([KS] 5.3); main result on how to solve the Dirichlet problem for the Poisson eq. in 3D using a Green's function ([KS] 5.4; a complementary derivation can be found in my lecture notes).
Lektion 6: Green's function for ODE problem (a simple example; it is written out in my lecture notes above); Green's function for the wave equation in 3D ([KS] 7.8; I used a different derivation computing the fundamental solution using 3D Fourier transformation). Computation of Green's functions for half spaces using the method of mirror images ("spegling"; [KS] 5.5 and 4.2). Green's function for heat equation ([KS] 5.8; this we discussed already in the main course, and I only recall this here).
END PART 1
130501: The exam on Part 1, together with a suggested solution, are now available here.
Note that there will be another chance to do the written exam on Part 1 in the August exam week.
Lektion 7: Lagrangian formalism, symmetries and conservation laws. Remarks on friction in classical and quantum mechanics. Reading for next time: Tong's lecture notes, Section 2.
Plan for next lectures: discussion of the material you read in Tong's lecture notes. You can ask me questions about it, and I will try to fill in examples and details.
Remark: Some more information on the Lindblad equestions that I mentioned can be found on Wikipedia, e.g.
Lektion 8: I discussed various issues in response to questions about Tong's compendium, Section 2, including: Noether's theorem and applications; applications of the relations "symmetry-conservation laws", including Kepler problem; Calogero-Sutherland-Moser systems. Reading for next time: Tong's lecture notes, Sections 2 and 4. Plan for next lectures: discussion of the material you read in Tong's lecture notes. You can ask me questions about it, and I will try to fill in examples and details. Remark: A concise introduction to Calogero-Sutherland model can be found on Scholarpedia.
Lektion 9: I discussed various issues in Tong's compendium, Section 2, including: non-uniqueness of Lagrangian ("adding total time derivative"); Lagrangian for a particle in electro-magnetic field and gauge invariance; small oscillations. Reading for next time: Tong's lecture notes, Section 4.
Lektion 10: I discussed various issues in Tong's compendium, Section 3, including: Hamiltonian formalism (generalities); examples (particle in electromagnetic field etc.); Poisson brackets; conservation laws.
Lektion 11: Canonical transformations (summary of important results); Hamilton-Jacobi equation: summary, and solution of the Kepler problem using the HJ equation. Relation of Hamilton-Jacobi equation and Schrödinger equation.
NOTE: Since there will be a colloquium on Thursday 15.15-16.15h the lecture is shifted to 16.15-18.00h (same place; this was suggested to me in class today and agreed upon by all students attending).
Lektion 12: (Plan) Theory of canonical transformations. Summary