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*–