Final review

• Help sessions scheduled for Dec. 8 and 9, 6:30 pm in MPHY 213

• Your hand-written notes allowed• No numbers, unless you want a problem

with numbers• Math formulas from Taylor will be given

How to prepare

• Review your lecture notes and make sure they are complete

• Solve your homework• Solve your mid-term tests• Solutions are posted, but don’t look at them before

you solve the problem!• Work out examples in textbook and lecture notes, and

look through end-of-chapter problems• Don’t hesitate to contact me if you have any difficulties

Math

• Vectors, dot and cross product• Polar, cylindrical, and spherical coordinates• Calculus

– Integrate by substitution of variable– Line element ds2 in standard coordinate systems

• Vector calculus (formulas will be given)• Differential equations:

– Solve by separation of variables– Solve linear equations by substitution x ~ exp(λt)– Apply initial conditions

• Approximations, expansions, linearization

Conservation laws: Know when and how to apply them

• Momentum• Angular momentum• Energy (potential energy, work-energy theorem)

• These quantities are additive• P and L are vectors; only some of their

components may be conserved

1D motion

• General solution for E = const• Periodic motion• Critical (equilibrium) points. Linearization!

Small oscillations around equilibrium! Phase plane!

Lagrangian mechanics

• Velocity and kinetic energy in cylindrical and spherical coordinates

• Euler-Lagrange equations and their general properties: – cyclic coordinates and integrals of motion – dropping total derivatives

• Similarity and virial theorem• Equilibrium points, linearization, small

oscillations!• Lagrangian for a particle in the EM field

Problem solving tips

• If you are not sure, choose Cartesian coordinates and then convert into any other coordinates

• Determine the number of degrees of freedom. Use constraints to eliminate extra variables

• Identify and drop total derivatives• Identify cyclic coordinates and use

corresponding integrals of motion instead of E-L equations

Blockbuster problems

• Particle on a sphere• Particle inside or outside a conical surface• Pendulum with movable suspension point• A bead on a (rotating) wire of certain

shape• Charge in constant electric and magnetic

fields

Central force

• Review chapter 8, LL chapter, class notes, and homework

• Conservation of E and L• Properties of orbits in a fixed central force potential• Effective radial motion and potential• Applying similarity and virial theorem • Orbits in a gravitational field. General formula p/r = 1 +

ecosφ. Energy and angular momentum of the orbit• Changing parameters, changing orbits, tangential boosts

Two-body problem

• Relationship between C.O.M. and lab frames. Relative motion, μ-point

• Lagrangian for the relative and COM motion. E-L equations

• Two particles interacting with a central force and in an external field

Collisions and scattering• C.o.m. and lab frames: conservation laws.

Relationship between c.o.m. and lab frames• Kinematic formulas for angles, velocities,

momenta etc. • Formulation of the scattering problem• Impact parameter, scattering angle, solid angle• Scattering cross-section in the c.o.m. and lab

frames (for incident particles and targets)

Special cases

• Coulomb scattering• Scattering by an elastic surface of

revolution• Capture by an attractive center and by a

finite-size object• Small-angle scattering

Flux of particles

• The flux density• The transfer equation• Mean free path, collision frequency,

attenuation coefficient, optical depth

Non-inertial reference frames

• Determine direction and magnitude of all forces• Write equations of motion in components and

solve it• Centrifugal and Coriolis force• Projectile motion on Earth

– Expansion in powers of Ω

• Motion on a rotating platform• Magnitude of tidal force• Roche limit