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Fall 2004: PHYS350 Applications of Classical Mechanics

Instructor: Philip Stamp Contact:
  • Office: Hennings 311A
  • office phone: 604-822-5711

Lectures: 10-11am on Tuesday & Thursday, in Hennings 201;
1.00-2.00 pm on Friday, in room D239 of Buchanan (PLEASE NOTE ROOM CHANGE).

Grading:

  • 40%: assignments
  • 60%: final exam

Practice mid-term questions

Practice final exam

UBC Calendar description: PHYS 350: Review of principles. Particle mechanics: Euler's equations, tops and gyroscopes, motion of the Earth, Lagrangian and Hamiltonian methods. Variational principles in optics and mechanics, Liouville's theorem and statistical mechanics. The relationship between classical and quantum mechanics. [3-0-0]

Prerequisite: PHYS 270.


Classical Mechanics is that part of physics which describes the motion of bodies according to Newton's laws, first written down in the 17trh century. However it can be formulated in a much more elegant and useful way using principles that were developed much later, mainly by Hamilton and Lagrange. The dynamical behaviour of particles, solid and liquid bodies, and gases then all derive from a single "Principle of Least Action"- a principle of such power and generality that it is still the source of all our dynamical laws, even in the 21st century of quantum field theory and gravity. The aim of this course is to give some idea of how this principle works, and then to show how it can be used in a very efficient way to solve practical problems in the statics and dynamics of moving and rotating bodies, including oscillating systems, and objects moving in central force fields.

The core of the course will deal with 4 topics, viz.,

  • (i)the Lagrangian formalism, its derivation from the principle of least action, and the relationship to the Hamiltonian formulation of mechanics. The main point here will be to give an appreciation of the mathematical structure and how it is connected to the physics, and to show you how one can systematically set up Lagrangians to solve problems.
  • (ii)The dynamics of oscillating systems, including coupled oscillators- these appear throughout Nature, in contexts ranging from simple mechanical and electrical systems to more complex objects like musical instruments, atoms and molecules, etc. This will cover important physical topics like resonance, and mathematical techniques like Laplace and Fourier transforms, and Green functions. Some attention will also be given to non-linear oscillators and non-linear dynamics.
  • (iii)The motion of rigid bodies- in which angular momentum plays a crucial role. Topics like precession and nutation, and rotational stability, will be covered, and applied to a wide variety of real systems (not just tops and gyroscopes!), mostly using the Euler equations and related techniques. If there is time, some continuum mechanics.
  • (iv)Motion in a central field, particularly in gravitational fields. This will cover orbital motion and stability, with excursions into planetary dynamics and other related astronomical phenomena, and non-linear dynamics.

I will repeatedly, in this course, show both the power and unified nature of the techniques by making diversions to look at interesting examples- often discussed with less than total rigour. These will include

  • the way in which the action principle can be used to derive the laws of optics, and to formulate quantum mechanics and field theory.
  • Interaction of non-linear oscillators, chaotic behaviour
  • Planetary dynamics, long-term stability, and the KAM theorem
  • Exotic astrophysical objects (eg., black holes)
  • Musical instruments

And any other topic that people might be particularly interested in.