F5510 Analytical Mechanics

Exercises: Wednesdays 09:00-10:00.
Lectures : Wednesdays 10:00-12:00.

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

Course Plan, Fall 2025:

Wednesday 17.09.25 at 09:00
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 24.09.25
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 01.10.25
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 08.10.25
Dictionary between Point Mechanics and Field Theory; Total derivative terms;
Exercises: [G3] 2.18 + 2.20;

Wednesday 15.10.25
Noether's Theorem, Symmetry, Local vs. Global, Conservation Law, On-shell vs. Off-shell [KB] [G3] 13.7; Special Relativistic Version of "Kinetic Term" T [LL2] 8;
Exercises: [G3] 2.3 + 2.12;

Wednesday 22.10.25
From Discrete to Continuous Indices, Longitudinal Motion in Wire [G3] 13.1;
Exercises: Handout: Maxwell Eqs. from Action Principle;

Wednesday 29.10.25
Legendre Transformation, Hamilton's Equations [G3] 8.1-8.2, 8.5 [LL1] 40;
Exercises: Handout: Geodesic Equation [LL2] 86-87;

Wednesday 05.11.25
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;
Constants of motion, Poisson's Theorem [G3] 9.6 [LL1] 42;
Exercises: [G3] 13.1;

Wednesday 12.11.25
Extended Canonical Transformations (CT), The Four Archetypes of CT [G3] 9.1 [LL1] 45;
Exercises: [G3] 8.23 + 8.28;

Wednesday 19.11.25
Symplectic Transformations (ST) [G3] 9.4;
Infinitesimal Canonical Transformations (ICT) [G3] 9.4;
Liouville Theorem [G3] 9.8 [LL1] 46;
Exercises: [G3] 8.2;

Wednesday 26.11.25
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 03.12.25
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 10.12.25
Hamiltonian Formulation of EM field; Exercises: Handout + [G3] 10.17;

Wednesday 17.12.25
Dirac-Bergmann Analysis; Primary, Secondary & Tertiary Constraints;

New! Tuesday 13.01.26
Consultation hour at 10:00 in F3 3015, 3rd floor, building 6.

Tentative Exam Schedule:

New! Thursday 15.01.26
Exam at 09:00 in FLenc 3028, 3rd floor, building 6.

09:00 Jakub
09:30 Alena
10:00 Adam
10:30 Lara
11:00 Lukas

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.

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 .