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 ds 2 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

Kinematics in polar coordinates y x  Circular motion:

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