Advanced Classical Mechanics
Advanced Classical Mechanics
Lecture notes (current status of lectures: see www.piazza.com)
In order to orient you about the scope and level of this class, these are left online from last year. The
order may change as we go on, but it is unlikely that we will have time to add more topics, despite
the desire to do so.
For those who want the chapter numbers from the book, here they are: all of Chs. 1, 2, 3; Ch. 7, Sects. 40,42,49; Ch. 6, skip Sects. 36,37,38.
pp.1-13: Review of the goals and scope of classical mechanics (1-4). Statement of the principle of extremal (“least”) action (5-8). Functionals, extremals, and the Euler-Lagrange equations from the least action principle (9-13).
pp.13.1-27: Two comments on variations (13.1-13.2). Equivalence of Newton to Euler-Lagrange (14). The shortest path on the plane via variational principle for length functional (14.1-14.2). Why does the variational principle work? - Modern understanding via the classical limit of the Feynman path integral of quantum mechanics (15-18). A brief “ode” to the usefulness of action principle (19). Where does the Lagrangian come from? - Galilean relativity principle and the form of the free-particle Lagrangian, as a first example of how symmetry principles determine form of Lagrangian (20-27).
pp. 20.1-20.2: On spacetime symmetries in classical nonrelativistic and relativistic mechanics.
pp. 27.1-27.3: For the curious - nonrelativistic free particle Lagrangian as a limit of the relativistic one.
pp.28-57: Lagrangian of a free particle in various coordinates (28); interaction Lagrangians (29-32); non-closed
systems (33); systems with constraints (34-36); time translation invariance and energy conservation (37-40); space translation invariance, momentum conservation, and applications (41-49); rotational invariance and angular momentum conservation (50-57). NOTE: A careful student pointed out a typo on p. 38, in the expression for the energy, on top line as well as 5th line from above: dL/d\dot{q} should be multiplied by \dot{q}, not q!
pp. d.1-86.3: The conserved quantity due to Gallilean invariance of action. Note one important lesson from this
example: Noether’s theorem, stating that conserved quantities exist, requires that the action (not necessarily the Lagrangian) be invariant under the symmetry transformation. Under Gallilean symmetry transformations, the Lagrangian transforms by a total derivative term. This affects the expression for the quantity conserved (due to Gallilean invariance) by the dynamical evolution as shown in these notes (d.1-d.2; please note a couple of obvious typos on these pages; also, the TA will discuss the topic during the Friday, Feb. 3 tutorial). Number of maximal possible integrals of motion (conserved quantities) (c.0); free particle and conserved quantities (c.1-c.2); central potential, its integrals of motion, and reduction to a one-dimensional problem (58-65 and c.3-c.4); “Kepler problem” via Rungle-Lenz vector (c.5, 86.1-86.3).
pp. 73-82: Euler’s theorem on homogeneous functions (73-74); homogeneous potentials and similar solutions (74-77);
examples of use and “virial theorem” (77-80); use of the virial theorem (81-82).
During the week of Feb. 13 (while I am away) Omar will discuss the promised pages d.1-d.2 as well as topics (precise choice will be his!) on motion in central potentials, both 1/r or otherwise. Some old notes of mine on this topic are
found on p. 66-72 in these notes and on p. 83-90 in these other older set of notes. You will notice that “scattering theory” appears on these notes, too - however, given our limited time, we have to prioritize, and I have decided that discussing Hamiltonian mechanics is more important (feel free to study scattering on your own, it is important, too, as it is a key concept applied in all kinds of physics, from solid state and astrophysics to particle physics). We will begin our study of Hamiltonian mechanics on Friday, Feb. 10, and continue after the break.
p. 157-169: Lagrangian and Hamiltonian mechanics - math preliminaries on Legendre transform (157-159); Hamiltonian as a Legendre transform of the Lagrangian and the Hamilton equations of motion, for one and many variables (160-163); conservation laws in Hamiltonian mechanics (164); involution property of Legendre transform and the Lagrangian (165); the notion of phase space and phase flow (165-166); the phase flow, Poisson brackets, and their use in the transition to quantum mechanics (``quantization”) (167-169). Apparently, my definition of Poisson brackets has opposite sign to that in Landau-Lifshitz, Goldstein and other classic books. But so long as I stick with it consistently, no problem should arise. Note that there will be no notes on adiabatic invariants, the Liouville theorem, and Poincare recurrence theorem - these will be only qualitatively described to stress the interesting connections to statistical physics and quantum mechanics. If you are not familiar with these subjects, don’t worry, this is purely “educational” discussion, i.e. it won’t be on the exam.
Prompted by a question asked by a student, a discussion of energy conservation in external electromagnetic fields.
p. 106-122: rigid bodies, their motion and angular velocity (106-109.1); independence of angular velocity of chosen
origin (110-111.1); kinetic energy of a rigid body (112-114); inertia tensor and principal moments of inertia (115-119);
parallel axes theorem (120-121); Lagrangian of rigid body (121); inertia tensor of a body with a continuous mass distribution (122).
p. 123-131: some examples - kinetic energy of a rolling cylinders with different mass distributions and the
instantaneous axis of rotation (123-126); momenta conjugate to angular velocities and the Hamiltonian
of a rigid body (126.1-126.3); the angular momentum and the tensor of inertia (127-129); parametrizing rotations using Euler angles (130-131).
p. 131.1-.9 and 131.14: the mathematics of rotations and Euler angles and their use to find the angular velocity of a body (see also Hw. 6, where the missing pages 131.10-131.13 will be filled!)
p. 132-139: using the Euler angles to study the motion of a free symmetrical top (132-135); equations of motion for a rigid body (136-139) [the heavy top will be discussed during last tutorial!]
“To those who study the progress of exact science, the common spinning-top is a symbol of the labours and the perplexities of men.”
James Clerk Maxwell
p. 144-150: the idea of “inertial forces” and motion in non-inertial frames (144); the Lagrangian in a non-inertial frame as a change of coordinates (145-148); the Euler-Lagrange equations in a non-inertial frame and the interpretation of the various “forces” (149); simple examples (150).
p. 151-164 [to be discussed during the final week of classes in the order given, time permitting]: the Hamiltonian in a non-inertial frame (151-152); rigid body motion + inertial forces = a qualitative discussion of the gyrocompass (157-158); similarities between inertial forces and gravity and an idea about the equivalence principle (153-156); on the “genesis” of the least action principle: Fermat’s principle (159-163); attempt to formulate the main “lesson” of this course (164).
Prof. Erich Poppitz
office: 1113A, 416-946-7546
email: poppitz physics utoronto ca
Syllabus and literature
This course introduces the principle of least action, Lagrangian mechanics, symmetries and conservation laws, central field motion, Euler angles, solid body motion, and motion in noninertial frames. Basic features of Hamiltonian dynamics are also discussed.
Required - “Mechanics” by L. Landau and I. Lifshitz
Recommended more advanced reading - “Mathematical methods of classical mechanics”
by V. Arnold.
Recommended somewhat less advanced reading - any book on classical mechanics you like. For a book with many problems, see, for example “Introduction to classical mechanics with problems and solutions” by D. Morin
As you will note, Landau and Lifshitz's book is a very condensed book and often in an one-hour class we will go through one or two pages only. My motivation for this choice is that it is the classic of the field---which is hard to beat in its completeness and depth (it is also thinner and hence easier on the wallet). I will be posting my class notes which will contain extended derivations of the covered material.
We will be using www.piazza.com for updates, posting of homework, solutions,
and online discussions. All registered students will receive an email inviting them to register.
The “competition”: MOOC’s? (I couldn’t find any on this advanced topic).
However, the Stanford School of Continuing Studies has posted 10 lectures on “Classical Mechanics” by Leonard Susskind, one of the world’s most prominent theoretical physicists, who has made many seminal contribution in elementary particle theory. These lectures last a total of 1040 mins = 20.8 x 50 mins, i.e., almost as much as our class of 24 50 min lectures. Check them out on this playlist (notice the steadily dropping number of views as the lectures progress - let’s try to avoid this with our actual class!). To spark your interest, in one of the lectures Prof. Susskind solves one of our homework problems on the board.
NOTE: The notes below are left here for reference. They contain unfixed errors, incl. undiscovered ones.
Thanks to Darryl Hoving, via xkcd, a very relevant cartoon: