Dynamics of a One-Dimensional Crystal
Michael Fowler
The Model
Notation! In this lecture, I use for the spring constant ( is a wave number) and  for frequency ( is a root of unity).
A good classical model for a crystal is to represent the atoms by balls held in place by light springs, representing valence bonds, between nearest neighbors. The simplest such crystal that has some realistic features is a single chain of connected identical atoms. To make the math easy, we'll connect the ends of the chain to make it a circle. This is called "imposing periodic boundary conditions". It is common practice in condensed matter theory, and makes little difference to the physics for a large system.
We'll take the rest positions of the atoms to be uniformly spaced,  apart, with the first atom at 
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Driven Oscillator
Michael Fowler (closely following Landau para 22)
Consider a one-dimensional simple harmonic oscillator with a variable external force acting, so the equation of motion is
which would come from the Lagrangian
(Landau "derives" this as the leading order non-constant term in a time-dependent external potential.)
The general solution of the differential equation is  , where  , the solution of the homogeneous equation, and  is some particular integral of the inhomogeneous equation.
An important case is that of a periodic driving force  A trial solution 
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Elastic Scattering
Michael Fowler
Billiard Balls
"Elastic" means no internal energy modes of the scatterer or of the scatteree are excited -- so total kinetic energy is conserved. As a simple first exercise, think of two billiard balls colliding. The best way to see it is in the center of mass frame of reference. If they're equal mass, they come in from opposite directions, scatter, then move off in opposite directions. In the early days of particle accelerators (before colliders) a beam of particles was directed at a stationary target. So, the frame in which one particle is initially at rest is called the lab frame. What happens if we shoot one billiard ball at another which is initially at rest? (We'll ignore possible internal energies, including spinning.) The answer is that they come off at right angles. This follows trivially from conservation of energy and momentum (in an obvious notation)
and Pythagoras' theorem.
Discovery of the Nucleus
The first significant use of scattering to learn about the internal structure of matter was Rutherford's use of 
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A Vectorial Approach: Hamilton's Equation and the Runge Lenz Vector
(Mainly following Milne, Vectorial Mechanics, p 235 on.)
Laplace and Hamilton developed a rather different approach to this inverse-square orbit problem, best expressed vectorially, and made a surprising discovery: even though conservation of angular momentum and of energy were enough to determine the motion completely, for the special case of an inverse-square central force, something else was conserved. So the system has another symmetry!
Hamilton's approach (actually vectorized by Gibbs) was to apply the operator 
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We'll assume that near the minimum, call it 
, the potential is well described by the leading second-order term, 
, so we're taking the zero of potential at 
assuming that the second derivative 
, and (for now) neglecting higher order terms.
origin to