F5510 Analytical Mechanics

Exercises: Wednesdays 14:00-15:00.
Lectures : Wednesdays 15:00-17:00.

Both lectures and exercises take place in FLenc, 3rd floor, building 6.

Course Plan, Fall 2024:

Wednesday 18.09.24 at 14:00
Cancelled due to floods.

Wednesday 25.09.24
Holonomic, Semi-Holonomic & Non-Holonomic Constraints [G3] 1.3;
Principle of Virtual Work, D'Alembert's Principle, From Newton's to Lagrange's Eqs. [G3] 1.4;
Applications, Atwood's Machine [G3] 1.6;

Wednesday 02.10.24
Gen. Potential for Lorentz Force [G3] 1.5;
Friction Forces, Rayleigh's Dissipative Function [G3] 1.5;
Canonical Momentum, Energy Function, Energy Conservation [G3] 2.6;
Virial Theorem [G3] 3.4;
Exercises: Handout: Two pendulum problems;

Wednesday 09.10.24
Gen. Potential for Fictitious Forces [LL1] 39;
Lagrange Eqs. with Semi-Holonomic Constraints [G3] 2.4;
Variational Derivative, Principle of Stationary/Least Action [G3] 2.1-2.3;

Wednesday 16.10.24
Dictionary between Point Mechanics and Field Theory; Total derivative terms;
Tensors, scalars, vectors, co-verctors & indices;
Exercises: [G3] 2.18 + 2.20;

Wednesday 23.10.24
Noether's Theorem, Symmetry, Local vs. Global, Conservation Law, On-shell vs. Off-shell [KB] [G3] 13.7;
Exercises: [G3] 2.3 + 2.12;

Wednesday 30.10.24
Special Relativistic Version of "Kinetic Term" T [LL2] 8; Total action for E&M;
Exercises: Handout: Maxwell Eqs. from Action Principle;

Wednesday 06.11.24
From Discrete to Continuous Indices, Longitudinal Motion in Wire [G3] 13.1;
Exercises: Handout: Geodesic Equation [LL2] 86-87;

Wednesday 13.11.24
Legendre Transformation, Hamilton's Equations [G3] 8.1-8.2, 8.5 [LL1] 40;
Symplectic Notation [G3] 9.4;
Poisson Bracket (PB) [G3] 9.4 [LL1] 42;
Classical Mechanics vs. Quantum Mechanics, Hamilton's EOM vs. Heisenberg EOM [G3] 9.5;
Exercises: [G3] 13.1;

Wednesday 20.11.24
Constants of motion, Poisson's Theorem [G3] 9.6 [LL1] 42;
Extended Canonical Transformations (CT), The Four Archetypes of CT [G3] 9.1 [LL1] 45;
Exercises: [G3] 8.23 + 8.28;

Wednesday 27.11.24
Symplectic Transformations (ST) [G3] 9.4;
Infinitesimal Canonical Transformations (ICT) [G3] 9.4;
Liouville Theorem [G3] 9.8 [LL1] 46;
General Poisson Bracket;
Exercises: [G3] 8.2;

Wednesday 04.12.24
Abbreviated Action; Maupertuis' Principle [G3] 8.6 [LL1] 44;
Hamilton-Jacobi Theory, Hamilton's Principal S fct. [G3] 10.1-10.2 [LL1] 47;
Hamilton's Characteristic W fct. [G3] 10.3;
Exercises: [G3] 9.8 + 9.25 + 9.31;

Wednesday 11.12.24
General Poisson Bracket, Darboux Coordinates, Darboux' Theorem; Symplectic 2-form;
Symplectic 2-form and Symplectic 1-form Potential;
Principle of least/extremal action and Hamilton's eqs. for general PB;
Exercises: [G3] 9.30 + 10.5;

Wednesday 18.12.24
Hamiltonian Formulation of EM field;
Exercises: Handout + [G3] 10.17;

New! Friday 07.02.25
Consultation hour at 10:00 in FLenc 3028, 3rd floor, building 6.

Tentative Exam Schedule:

New! Monday 10.02.25
Exam at 08:00 in FLenc 3028, 3rd floor, building 6.

08:00 Denisa
08:30 Daniel
09:00 Tomas
09:30 Petr
10:00 Jachym
10:30 Jan R
11:00 Filip
11:30 Jindrich
12:00 Jan H

NB: Due to new regulation that 3 people should be present during exam, I ask that each student after their exam has finished stay listening to the next student.

List of Exam Questions:

1. From Newton's to Lagrange's Eqs., D'Alembert's Principle, Principle of Virtual Work [G3] 1.4;
2. Gen. Potential for Lorentz Force [G3] 1.5;
3. Canonical momentum, Energy Function h, Energy Conservation [G3] 2.6, Virial Theorem [G3] 3.4;
4. Variational Derivative, Principle of Least/Extremal Action [G3] 2.1-2.3;
5. Noether's Theorem, Symmetry, Local vs. Global, Conservation Law, On-shell vs. Off-shell [KB] [G3] 13.7;
6. From Discrete to Continuous Indices, Longitudinal Motion in Wire [G3] 13.1;
7. Maxwell Eqs. from Action Principle [LL2] 23,27,30;
8. Legendre Transformation, Hamilton's Equations [G3] 8.1-8.2, 8.5 [LL1] 40;
9. Poisson Brackets, Poisson's Theorem, (PB) [G3] 9.4 [LL1] 42;
10. Canonical Transformations (CT), The Four Archetypes of CT [G3] 9.1 [LL1] 45;
11. Infinitesimal Canonical Transformations (ICT) [G3] 9.3;
12. Hamilton-Jacobi Theory, Hamilton's Principal S fct. [G3] 10.1-2 [LL1] 47;
13. Hamilton-Jacobi Theory, Hamilton's Characteristic W fct. [G3] 10.3;

Course Material:

[G2]: Herbert Goldstein, "Classical Mechanics", Eds. 2.
[G3]: Herbert Goldstein, "Classical Mechanics", Eds. 3. (Click here for a list of corrections).
[LL1]: Landau and Lifshitz, Vol. 1, "Mechanics".
[LL2]: Landau and Lifshitz, Vol. 2, "The Classical Theory of Fields".
[KB]: Klaus Bering, "Noether's Theorem for a Fixed Region", arXiv:0911.0169 .

Nota Bene:

1. All references to [LL2] should be considered supplementary reading, as the presentation during the lectures differs substantially.
2. What Goldstein [G3] calls "Hamilton's principle" is usually called the "principle of stationary/least action".
3. What Goldstein [G3] calls "principle of least action" [G3] 8.6, is usually called the "principle of abbreviated action" or "Maupertuis' principle".
4. Note that Poisson brackets in [LL1] have the opposite sign convention.
5. The treatment of Lagrange equations for semi-holonomic & non-holonomic constraints in [G3] 2.4 is inconsistent with Newton's laws, and has been retracted on [G3]'s errata homepage . For more info, see also M.R. Flannery, "The enigma of nonholonomic constraints", Am. J. Phys. 73 (2005) 265 .

Supplementary References:

[JS]: J.V. Jose and E.J. Saletan, "Classical Dynamics: A Contemporary Approach", 1998.
[L]: N.A. Lemos, "Analytical Mechanics", 2018.
[M]: P. Mann, "Lagrangian and Hamiltonian dynamics", 2018.
[SWM]: G.J. Sussman and J. Wisdom with M.E. Mayer, "Structure and Interpretation of Classical Mechanics", html .