FK7049 - Analytical Mechanics VT20

On this page you find information about the course "Analytical Mechanics" (kurskod FK7049) given during spring 2020 for students at the physics bachelor/master program at Stockholm university.

Course content

The aim of the course is to give an understanding of analytical mechanics, and to be able to understand its role as a background for quantum mechanics. The course can be devided into five parts:

Lagrangian theory: Constraints and generalized coordinates, d'Alembert's and Hamilton's principles, Lagrange and Euler-Lagrange equations, variational methods, constraints and Lagrange multipliers, conservation laws.

Hamilton theory: Legendre transforms, canonical transformations, Generating functions, Phase space, Noether's theorem, Liouville's theorem.

Hamilton-Jacobi Theory: The Hamilton-Jacobi equation, Hamilton's principal and characteristic functions, Action-angle variables.

Classical chaos: Lyapunov exponents, the KAM theorem, Poincaré sections.

Connection to quantum mechanics: Poisson brackets.

Course goal

After the course, the student is expected to have understood the basic ideas of analytical mechanics. That includes among others:

Understand the idea of generalized coordinates and how they are connected to contraints. Understand how the Lagrange equations comes about.

Understand the connection between the Lagrange and the Hamilton formalism. Understand the connection between symmetries and conserved quantities.

Understand the ideas behind canonical transformations and generating functions.

Understand the connections between classical and quantum mechanics. Know the basics of Poisson brackets.

Understand the basics of Hamilton-Jacobi theory. Know the ideas behind action-angle variables.

Understand some basic principles of classical chaos; Lyapunov exponets and the KAM theorem.

Literature and teaching form

The course follows selected parts of the book:

H. Goldstein, C. Poole and J. Safko, Classical Mechanics" , 3rd edition, Pearson Education.

This book serves as the course literature. Earlier years, Ingemar Bengtsson's lecture notes Notes on Analytical Mechanics, lecture notes were used. We will steal a few problems from them, plus that I recommend that you read a few parts of his notes (see reading instructions).

The course consists of 15 lectures and 6 tutorials, see scheme below. Lectures will be given by Jonas Larson (jolarson@fysik.su.se) and the tutorials are led by Marcus Högås (marcus.hogas@fysik.su.se). Apart from these, the course contains also 3 occasions where we discuss the individual assignment topics.

Examination

The examination consists of a written exam plus one mandatory individual assignment.

The grading criteria for the assignment are found here. The assignment consists of three parts:

Part I - Written report. The student picks one of the assignments (see further down on this page for the different assignments). Perform a self-study and writes a report. The report should preferably be 3-5 pages (only the first 5 pages will be graded, so better keep it at most 5 pages).

Part II - Peer reviewing. The student reads two other reports and give written feedback on them.

Part III - Discussion. In class we discuss the various topics of the assignments. For a given topic, the students who studied this particular assignment is supposed to be more active in the discussion.

Each student can collect a maximum 4 points; 2.5 on the written report and 1.5 on the peer reviewing, while the discussion does not givepoints but is mandatory. These points are added to the total points collected on the written exam. NOTE! Each student need to do this assignment, it is not possible to skip it thereby and score zero bonus points!

Grading criteria

The course grading criteria can be downloaded here (Swedish) or here (English).

Course schedule

The schedule can be found here.

Lectures

The table below gives a most preliminary schedule of what the lectures and tutorials are planed to cover.

Date Topic

Lecture 1 Tues 21/1 Repetition of Newtonian mechanics, constrains, generalized coordinates.

Lecture 2 Wed 22/1 Generalized coordinates contin, d'Alembert's principle.

Lecture 3 Fri 24/1 Variatonal methods.

Lecture 4 Tues 28/1 Hamilton's principle, constrains and Lagrange multipliers.

Lecture 5 Wed 29/1 Conservation laws, Noether's theorem.

Tutorial 1 Fri 31/1 The Lagrangian, Lagrange's equations.

Lecture 6 Tues 4/2 Hamilton's equations, Legendre transformations.

Lecture 7 Wed 5/2 Canonical transformations.

Tutorial 2 Fri 7/2 Variational methods, Legendre transforms.

Lecture 8 Tues 11/2 Cannonical transformations continued. Poisson brackets.

Lecture 9 Wed 12/2 Poison brackets, connections to quantum mechanics.

Tutorial 3 Fri 14/2 Hamilton's equations, symmetries.

Lecture 10 Tues 18/2 Phase space, Liouville's theorem.

Lecture 11 Wed 19/2 Hamilton-Jacobi theory.

Tutorial 4 Fri 21/2 Canonical transformations.

Lecture 12 Tues 25/2 Hamilton-Jacobi theory.

Lecture 13 Wed 27/2 Central forces, moments of inertia.

Tutorial 5 Fri 28/2 Canonical transformations and Hamilton-Jacobi theory.

Lecture 14 Tues 3/3 Inertia tensor contin. Classical chaos, Lyapunov exponents.

Lecture 15 Wed 4/2 Classical chaos, KAM theorem, Poincaré sections.

Tutorial 6 Fri 6/3 Old exam.

`Discussion' 1 Tues 10/3 Common discussion about the individual assignments I.

`Discussion' 2 Wed 11/3 Common discussion about the individual assignments II.

`Discussion' 3 Fri 13/3 Common discussion about the individual assignments III.

Exam Fri 20/3 Everything

	Date	Topic
Lecture 1	Tues 21/1	Repetition of Newtonian mechanics, constrains, generalized coordinates.
Lecture 2	Wed 22/1	Generalized coordinates contin, d'Alembert's principle.
Lecture 3	Fri 24/1	Variatonal methods.
Lecture 4	Tues 28/1	Hamilton's principle, constrains and Lagrange multipliers.
Lecture 5	Wed 29/1	Conservation laws, Noether's theorem.
Tutorial 1	Fri 31/1	The Lagrangian, Lagrange's equations.
Lecture 6	Tues 4/2	Hamilton's equations, Legendre transformations.
Lecture 7	Wed 5/2	Canonical transformations.
Tutorial 2	Fri 7/2	Variational methods, Legendre transforms.
Lecture 8	Tues 11/2	Cannonical transformations continued. Poisson brackets.
Lecture 9	Wed 12/2	Poison brackets, connections to quantum mechanics.
Tutorial 3	Fri 14/2	Hamilton's equations, symmetries.
Lecture 10	Tues 18/2	Phase space, Liouville's theorem.
Lecture 11	Wed 19/2	Hamilton-Jacobi theory.
Tutorial 4	Fri 21/2	Canonical transformations.
Lecture 12	Tues 25/2	Hamilton-Jacobi theory.
Lecture 13	Wed 27/2	Central forces, moments of inertia.
Tutorial 5	Fri 28/2	Canonical transformations and Hamilton-Jacobi theory.
Lecture 14	Tues 3/3	Inertia tensor contin. Classical chaos, Lyapunov exponents.
Lecture 15	Wed 4/2	Classical chaos, KAM theorem, Poincaré sections.
Tutorial 6	Fri 6/3	Old exam.
`Discussion' 1	Tues 10/3	Common discussion about the individual assignments I.
`Discussion' 2	Wed 11/3	Common discussion about the individual assignments II.
`Discussion' 3	Fri 13/3	Common discussion about the individual assignments III.
Exam	Fri 20/3	Everything

Reading instructions

The main content of the couse follows Goldstein's book, but a few parts are taken from Ingemar's lecture notes (find them here). To get a more detailed idea which parts of the book and the notes that are included in the course, have a look at the reading instructions here.