Energy ¨ dynamical matrix ¨ normal modes
Contact energy V  for particles i and j:
V = 1/2 (1 - r)2
separation
particle diameters
r < 1
Éexpressed in terms of displacement dRi , dRj, this gives energy
dynamical matrix depends on contact geometry
dE = dR| M |dR
vector of all 3N displacements dRi
Eigenstates of M are the normal modes;
eigenvalues are squared frequencies w2 (for particles of mass 1)
Variational bound: for any displacement field |dR* with dR*| 1 |dR* = 1
 lowest eigenvalue w02 £ dR*| M |dR*