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
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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