PHYS 350 Learning Outcomes
Topics marked with * are covered in prerequisite courses
Topics marked with ** are optional
Theory:
- Write down a Lagrangian in minimal generalized coordinates
- Change variables in a given Lagrangian
- Obtain euler-lagrange equation, conserved momenta and energy (if conserved)
- * solve the equations of motion; obtain quantities of interest
- explain how a variational derivative can be used to solve optimization problems
involving unknown functions; give examples.
- take a variational derivative given an action functional to obtain a
differential equation that is satisfied at the extremal point of the action
- ** obtain a conserved quantity associated with a symmetry of the action
- Obtain a Hamiltonian given a Lagrangian
- Find poisson brackets of any two observables
- Understand how poisson brackets can be used to write flow
equations (for a flow induced by any observable); what are the consequences of a
poisson bracket vanishing; what it means if a poisson bracket of two
observables: vanishes and does not vanish.
- Draw simple flows in (2d) phase space describing evolution due to an observable
(including hamiltonian)
- * find time-evolution of a hamiltonian system by combining conserved quantities
and hamilonian equations
Applications:
1d motion
-
find turnaround points and explain their significance
-
know how to use integrals between the turn around points to compute
quantities such as period
central potential
-
recognize central potential problems
-
know how to use conservation laws relevant to the central force problem
-
simplify the two body problem to a 1d effective problem in the radial
direction
-
know how to use integrals between the turn around points to compute
quantities such as period and precession angle
-
understand orbital parameters and be able to connect them to energy,
angular momentum and other quantities
-
explain and use kepler's laws
rigid body
-
* solve mechanics problems involving rotations in 2d
- know the definitions of inertial tensor, angular momentum, angular
velocity and torque in 3d and use them to state the 2rd law of rotations
- understand the connection between rigid body rotation matrices and
angular velocity
- understand and use pricipal axes and the corresponding moments of inertia
be able to apply euler's equations to problems involving fixed axes
rotations, precessions and gyroscope motion, when appropriate
- be able to to use euler angles in the lagrangian approach to solve
problems involving fixed axes rotations, precessions and gyroscope motion, when appropriate
vibrations
-
* understand the general solution to 1d harmonic oscillator and be able to
write down specific solutions involving particular initial conditions
- find static solutions given a multidimentional potential
- expand the lagrangian to quadratic order near the static solution
- find normal mode frequencies and the normal modes in a quadratic
lagrangian
- write down general and particular solutions for multidimensional harmonic
oscillations