These are the key concepts you should understand and on which you should
be able to perform related problems on the final exam. Many of the first items mimic the midterm study guide.
- Understand polar coordinate constructions and basis vectors.
- Whether linear or quadratic drag, be able to pull the terminal velocity
out of the differential equation, and be able to solve the differential
equation in one dimension (horizontal or vertical, linear or quadratic) to
get x(t) or y(t) and match to initial
conditions to replace the constants of integration.
- Understand complex exponential notation, Euler's formula, etc.
- Be able to compute integral properties of solid bodies, such as volume,
center of mass, moment of inertia, etc.
- Get yourself straight on angular motion: angular momentum, torque,
moment of inertia, and how all of these are related (e.g.,
L-dot = torque; L = Iω, etc.).
- Develop a reflex for invoking conservation of energy and momentum.
- Be familiar with the grad/del/nabla doohicky and its application in
constructing and evaluating conservative forces (e.g., grad-U and
- Given total energy, E and potential U(x), be
able to construct a differential equation for the motion that can be solved
to arrive at x(t).
- Be able to evaluate the stability of a system based on the potential
energy as a function of some (not necessarily Cartesian) coordinate or
- Given the differential equation for a damped oscillator, be able to
walk through its solution for the over- and under-damped cases, and match
the ultimate solution for x(t) to initial conditions.
- Understand how to cope with a sinusoidal driving force in a damped
oscillator and how to combine particular and homogeneous solutions to get
the general solution. Only the particular solution matters in the long
run, in the damped oscillator case.
- Be familiar with the resonance phenomenon: the amplitude peak and phase
shift. I would not require you to memorize formulas, but be able to
recognize and work with them if needed.
- Review the calculus of variations, and how to minimize some integral
along a path by clever use of the Euler-Lagrange equations. Often you have
a choice between paths described by y(x) or
x(y). The one with fewer derivatives to perform will
make for an easier solution (if the function has no y dependence,
then pick the y(x) version).
- At this point, you should be comfortable constructing a Lagrangian in
Cartesian or polar coordinates, and turning this into equations of motion
for the relevant variables.
- We'll leave the undetermined multiplier part for investigating
constraints off the exam. But don't forget about it entirely, because it
will crop up again...
- Review the technique by which stability is assessed from the Lagrangian
formulation: first and second derivatives of the variable are both zero at
an equilibrium point. Stability is evaluated by seeing if a small
displacement from equilibrium (with velocity still zero) makes the second
derivative oppose the displacement—in which case it's stable.
- Be able to retrace the development of the center of mass and reduced
mass formulation for two masses at r1 and
r2. Be able to go back and forth between these two and
the CoM pair R and r.
- Reduced mass problems often productively start from the reduced
mass/CoM form of kinetic energy:
T = ½M(R-dot)² +
½μ(r-dot)². Be able to derive this if
asked, or to just use it when needed.
- Be familiar with Ueff — how to construct it,
how to identify rmin and rmax given
some energy, and how to spot circular, elliptical, parabolic, and
hyperbolic orbits in the case of a 1/r potential.
- Be intimately familiar with the polar equation for orbits in a
1/r potential, which works for circles, ellipses, parabolas, and
hyperbolas. That is:
r = s/(1 + εcosφ). Lots
of investigations in class started with this equation, so review the ways
in which this relationship is used.
- Be familiar with ellipse geometry and interrelationships between:
s, a, b, c, ε,
rmin, rmax, etc.
- Be able to compute the angular momentum and energy for a particle given
some position and velocity relative to an attracting body, so that you may
identify and use (provided) formulas for determining orbital parameters.
Also be able to identify and use appropriate relationships for a,
s, energy, and Kepler's third law in general form for bounded
- Be able to follow/reproduce the development of a coupled oscillation
problem in two variables. How do the mass and spring matrices arise, and
how do we arrive at the relation
(K − ω²M)a = 0?
How do we solve for the normal mode frequencies and then find the normal
- Be able to express a mass matrix and spring matrix out of the kinetic
and potential energies of a two or three coordinate system, truncating to
second order as necessary (as in the double pendulum example, and in
section 11.5). Be able to solve for normal mode frequencies and motions.