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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 vel145@gmail.com or in person.

Link to the timetable

Tutorials are given jointly with Robert Murtagh murtagr@tcd.ie

Compulsory homeworks are checked by Eamonn O'Shea eaoshea@tcd.ie

Assessment

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.

Curriculum

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

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) , Solutions/Hints:
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) ,
Solutions
Tutorial #6 (Tue 11 Mar, Wed 12 Mar) ,
Solution for Part I,
Tutorial #7 (Tue 1 Apr, Wed 2 Apr) ,

Lecture notes

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.
Continuous systems

Home Work

***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.

Compulsory HW's

#	Topic	Due
1	Basics of Hamiltonian mechanics	7 Feb
3	Vector flows	21 Feb
5	Canonical transformations and Hamilton-Jacobi equation	7 Mar
7	Special relativity	26 Mar
9	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

Facultative HW's

#	Topic	Due	PDF	Solution
2	Exterior algebra and differential forms	12 Feb

Literature

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.

Mathematics: 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.

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
Vector flows. 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
Canonical transformations. 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