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