Fundamental physics, year 1 (Teknisk fysik, årskurs 4)

Advanced classical physics — Lecture notes

by Christian von Schultz

Lecturer: Per Salomonson

The course home page can be found at http://fy.chalmers.se/~tfeps/acp.dir/.

Tuesday 2008-09-02

Repetition of the Newtonian mechanics. Position, force, momentum. Particle systems. The weak form of the third law and the strong form of the third law. Conservation of momentum and angular momentum. Energy of a particle. Three equivalent criteria for conservative forces. Potential for particle systems. Centre of mass coordinates. The angular momentum of a particle system equals the angular momentum of the mass centre, plus the angular momentum due to motion relative to the centre of mass. The kinetic energy of a particle system equals the kinetic energy of the mass centre, plus the kinetic energy due to motion relative to the centre of mass. Rigid bodies, and the inertia tensor I_{a b}.

Friday 2008-09-05

Generalised coordinates, generalised force, generalised momentum. Lagrange's equations without a potential. Lagrange's equation for a conservative force, with a potential. Examples: planetary motion in polar coordinates (r, θ); plane pendulum. Holonomic constraints. Constraint forces and d'Alembert's principle.

Monday 2008-09-08

Holonomic constraints. Constraint forces and d'Alembert's principle. Elimination of the constraints from the set of equations (example: spherical pendulum). Lagrange multiplier method. Lagrangian for Lorentz force.

Tuesday 2008-09-09

No lecture notes today, sorry.

Friday 2008-09-12

Variational calculus. Soap bubble surface (minimisation of area). The analogy between variational calculus and ordinary differential calculus. Hamilton's principle. The action. Variational calculus with Lagrange parameters. Boundary terms and boundary conditions.

Monday 2008-09-15

For each conserved quantity there corresponds a continuous symmetry. The summation convention. Energy conservation — time translation. Momentum conservation — spatial translation. Conservation of angular momentum — rotation. Cyclic variables (ignorable variables). Elimination of cyclic coordinates from the Lagrangian. Jacobi's principle.

Tuesday 2008-09-16

Hamilton's equations. Going from a Lagrangian to the corresponding Hamiltonian. The action in the Hamiltonian formulation. Cyclic variables in the Hamiltonian formulation. Configuration space versus phase space. Flow in phase space. Liouville's theorem: “The phase fluid is incompressible.” Extended phase space.

Friday 2008-09-19

Canonical transformations. Point transformations. The generating function of a canonical transformation. Legendre transformation. Infinitesimal canonical transformations. Dynamical variables — infinitesimal canonical transformations. Poisson brackets. The Hamiltonian is the generator of infinitesimal time translation. Momentum as a generator of translation. Angular momentum as a generator of rotations. Symmetry transformation if the Hamiltonian is invariant. The Jacobi identity. The connection to quantum mechanics.

Monday 2008-09-22

Relativistic mechanics. Lorentz transformation. The metric η. Making theory relativistic = choosing equations covariant. The action from scalar pieces. The action for a free particle. The action for electromagnetism. Examples of a scalar field, a vector field and a tensor field. Curved spacetime. The Hamiltonian formulation in relativistic mechanics. Gauge transformations, gauge symmetry, gauge invariance, gauge theories.

Tuesday 2008-09-23

Exercise class.

Friday 2008-09-26

Mechanics of fields. The correspondence between particle theory and field theory. Equations of motion. Boundary conditions. Scalar field in relativistic theory [the Klein-Gordon Lagrangian]. Interpretation as a quantum field theory. The Hamiltonian formulation of field mechanics. The Poisson bracket. Current conservation and conservation of charge. Space translation and energy-momentum conservation.

Monday 2008-09-29

Integrable systems. Solving a system of linear differential equations with constant coefficients. System of free scalar fields. Lagrangian with O(n) symmetry. The action for the electromagnetic field.

Tuesday 2008-09-30

The Hamiltonian for the electromagnetic field. Gauge transformation of the electromagnetic potential A_μ. Field degrees of freedom. Second class constraints, first class constraints.

Friday 2008-10-03

The procedure of gauging. Gauging the O(n) symmetry (making the O(n) symmetry local). Covariant derivative. Gauge field potential. Gauge field strength. An O(n) Yang-Mills theory. The mathematical definition of a group.

Monday 2008-10-06

Symmetry groups. U(n) and O(n). SU(n) and SO(n). U(1) = SO(2). How the Lagrangian of the standard model is built. Goldstone's theorem and the Higgs mechanism.

Tuesday 2008-10-07

Solitions. O(1) model with spontaneous symmetry breaking [Wikipedia calls this a φ⁴ theory].

Friday 2008-10-10

No lecture notes today, sorry. I was holding a lecture myself, teaching the first years about ISO 31 and related matters.

Monday 2008-10-13

Tensors (continued). Addition. Outer multiplication. Contraction. Symmetrisation. Quotient rule (important for showing that a given object is a tensor). The Kronecker symbol — an invariant tensor. Metric manifold. Differentiation of a (p, q) field object. Tensor densities. Group theory: Subgroup. Invariant subgroup. Conjugacy. Equivalence relations: transitivity, symmetry, reflexivity. Equivalence classes (conjugacy classes). Albelian groups. Lie groups.

Friday 2008-10-17

Lie group. Lie algebra. Lie bracket. Adjoint representation. Ideal. Simple Lie algebra. Albelian ideal. Semisimple Lie algebra.

A word of thanks goes to Christer Samuelsson, who reported in some typos.