MA2342, advanced classical mechanics II, Hilary term 2014
This page is being continuously updated with all relevant information. For any questions and suggestions contact me
by e-mail email@example.com or in person.
Link to the timetable
Tutorials are given jointly with Robert Murtagh firstname.lastname@example.org
Compulsory homeworks are checked by Eamonn O'Shea email@example.com
Continuous assessment will contribute 20% to your final score. It includes 15% for home works and 5% for a multiple-choice test at the end of the term.
- Mathematical background: material from MA1214, MA2322, MA2332 (group theory, calculus on manifolds, differential equations) will be extensively used.
- The canonical equations
- Legendre transform
- Hamiltonian formalism
- Liouville theorem
- Canonical transformations
- Hamilton-Jacobi equations, action-angle variables
- Hamiltonian mechanics and calculus on manifolds
- Liouville integrability
- Special relativity
- Mechanics of continuous systems
Answers to MCT
Tutorials is an important part of this course. Some topics are covered only during tutorials, not during lectures. Moreover, some of the lectures will be intentionally replaced by tutorials to give you more time for exercises. I strongly encourage you to attend all the tutorials.
- John and Mary problem
- Questions from the first lecture
- Tutorial #1 (Thu 16 Jan, Mon 20 Jan, Thu 23 Jan)
- Tutorial #2 (Fri 24 Jan, Thu 30 Jan)
solutions and discussion:
- Comments on Levi-Civita symbol
- Tutorial #3 (Mon 3 Feb, Tue 4 Feb, Wed 5 Feb, Mon 10 Feb, Tue 11 Feb)
Solution for Part I,
Solution for part 2
- Tutorial #4 (Mon 17 Feb, Tue 18 Feb)
Solutions (also notes on canonical transformations )
- Scrodinger papers can be found online, google "Collected Papers on Wave Mechanics". For instance the following link works
- Tutorial #5 (Tue 4 Mar)
- Tutorial #6 (Tue 11 Mar, Wed 12 Mar)
Solution for Part I,
- Tutorial #7 (Tue 1 Apr, Wed 2 Apr)
Based on your request, here I upload lecture notes. The files below do not represent a complete account of lectures. I recommend Goldstein for an additional reading. Solving and studying solutions to the problems is a more important part of the course.
Dual vector space and differential 1-forms
Legendre transform, Hamiltonian equations of motion
Phase portraits (2d case)
Introducing symplectic structure
Continuity equation, Liouville's theorem (for phase volume), generalised Liouville's theorem (for symplectic structure)
Canonical transformation (typed notes)
Action as a function of coordinates, Hamilton-Jacobi equation
Special relativity, introduction
Minkowski space, Interval
General covariance, Light as solution of Maxwell equations, Maxwell equations in a covariant form
For a particle in EM field (action, equations of motion etc.) look also tutorial 7. Summary of this topic is in sections 1-3 of these notes . Section 4 there is also important, but definitely it will be not the part of examination. Note: you will have Maxwell equations next year with Tristan McLoughlin on a more profound level.
***Attach this cover to each homework:
Cover with instructions:
Cover without instructions: ***
- N best-marked HW's will contribute to the final score, where N is the number of submitted compulsory HW's.
- Each compulsory HW is worth 10 points, some of the facultative HW's can be worth as much as 15 points. If you get more than 50 points, your final score will be 50 points. Hence, by doing facultative HW's, you increase your chances to get the maximal score.
- I strongly suggest to do facultative HW for those who is determined to continue his career in theoretical physics or mathematics. Those of you who wish to receive an additional incentive, e-mail me and request to make the facultative HW compulsory. This will irreversibly change the evaluation rule for you: all 10 HW's will be contributing to the final score, with factor 1/2 each.
I advice you to start doing HW's well in advance, you might be not capable to finish everything overnight. The submission deadline will be always adjusted to be at least one week after you receive the necessary information on lectures or tutorials, but there are always questions which are feasible earlier than that. Assignments will appear online approximately two weeks before the deadline.
Though not required, it is a useful practice to take a look on the exercises and try to solve them even before the corresponding lecture or tutorial. The necessary literature links are provided in the HW's text.
||  Basics of Hamiltonian mechanics
||  7 Feb
|| Vector flows
||  Canonical transformations and Hamilton-Jacobi equation
||  7 Mar
||  Special relativity
||  26 Mar
||  Continuous systems
||  4 Apr
Mathematica nb for HW5:
Due to lack of time, I could not support properly facultative HW's this term. As a result, only one facultative HW was proposed
||Exterior algebra and differential forms
Classical mechanics is a well-established topic, there are dozens of good quality books which you may want to use. Below are the books that I am using to prepare lectures and tutorials.
- Wikipedia, youtube, research engines (usually information online is very nice, but be critical about what you are reading!)
- L.D. Landau and E.M. Lifshitz, Mechanics , Butterworth-Heinemann
- Herbert Goldstein, Classical Mechanics , third edition, Addison Wesley
- V.I. Arnold, Mathematical Methods of Classical Mechanics , Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Topics for exam
Exam will be 70%-80% of practical questions, and 20%-30% of theory.
For the theoretical questions, I will ask only those ones that were explained during lectures. I uploaded most of the lecture notes online, however these notes lack explanatory text. The best additional reading is Goldstein. Landau-Lifshitz is good for what concerns topics form the first half of the course.
For the practical questions, they will be very similar, though not fully equivalent, to those asked in tutorials and home works.
- The best and very fruitful way to prepare is to study solutions to home works and tutorials.
Dual vector space. Differential 1-forms and tangent vectors, their transformation when changing coordinate system, integrating differential 1-forms. Exact form, how to check that the form is exact. Legendre transform. Levi-Civita symbol. Polar, Spherical, and complex coordinates.
Tutorial 1, comments on Levi-Civita after tutorial 2.
No theoretical questions will be asked about purely mathematical topics, neither purely mathematical exercises will be proposed. But the mentioned topics might be a language used during solution of various exam questions.
Duration of exam is 2 hours. Exam will consist of 4 questions. You should solve 3 of your choice to get the full score.
2 questions will be devoted to the topics from:
- Hamiltonian mechanics.
Generalised co-ordinates and momenta. Hamiltonian as a Legendre transform of Lagrangian (theory and explicit computations). Lagrangians and Hamiltonians in different coordinate systems (polar, spherical, ...). Hamiltonian equations of motion. Poisson bracket, its basic properties. Conserved quantities.
Tutorial 2, HW 1
Generic vector flows. Gradient and symplectic vector flows (definition, how to check what type the vector flow is). Symplectic structure. Symplectic structure vs metric. Phase portraits. Classification of phase portraits for linear systems in 2d. Phase portraits for Hamiltonian systems (including non-linear cases). Continuity equation. Liouville theorem about conservation of the phase volume. Liouville theorem about conservation of symplectic structure.
Tutorial 3, HW 3
Definition of the canonical transformation. Verifying whether transformation is canonical. 4 types of generating functions. Action as a function of coordinates and time. Action as a generating function of the canonical transformations. Time-dependent canonical transformations.
Tutorial 4, HW 5
Hamilton-Jacobi equation. Derivation of HJ equation. Time-dependent and time-independent cases. Method of separation of variables (basics). Simple wave equations, dispersion relation. Group and phase velocity. HJ as an eikonal approximation of a wave equation, constant phase surfaces, particle movement in this language. Relation to the time-independent Schrodinger equation.
Won't be on exam: Relation to the time-dependent Schrondinger equation
Tutorial 5, HW 5
2 questions will be devoted to the topics from:
Basics of special relativity.
Einstein's principle of relativity. Distance contraction, time dilation. Interval. Space-like, time-like, light-like intervals and trajectories. Minkowski metric. 4-vectors. Lorentz transformations. Contravariant and covariant objects, tensors. Structure of the Lorentz group (boosts, rotations, discrete transformations). Minkowski diagrams. Absolute future and past, and "elsewhere". Addition of velocities formula. Rapidities. Paradoxes (twin, ladder in the house, and similar, with detailed explanations). Lagrangian, equations of motion, momentum, energy, for relativistic free particle.
Tutorial 6, HW 7
Derivation of a continuous system as a limit of a discrete one. Euler-Lagrange equations of motion, finding some simple solutions (like in HW). Stress-Energy tensor, related conservation laws.
- Electromagnetic field.
Light as a solution of the Maxwell equations. Vector potential, electromagnetic tensor. Covariant formulation of the Maxwell equations. Continuity equation for 4-current as a consequence. Actions: for a free relativistic particle, its interaction with EM field, for free EM field. Deriving equations of motions from these actions, i.e. for motion of a particle in EM field and a pair of Maxwell equations. Rewriting the obtained equations back in the 3-dimensional notations.
Example of exam
Profit from having an example! Do not look the sample exam questions immediately. First study solutions to HW's, tutorials, and a bit of theory. Then allocate 2 hours and give it a try.
Solution (by Jean Lagacé and Adam Keilthy, with corrections)