1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 53

interacting with a static electromagnetic field is given by

H=

∑N

i=1

(pi−qiA(ri)/c)^2

2 mi

+

∑N

i=1

qiφ(ri) +

N∑− 1

i=1

∑N

j=i+1

qiqj

|ri−rj|

,

whereA(r) andφ(r) are the vector and scalar potentials of the field, respec-

tively, andcis the speed of light. In terms of these quantities, the electric and

magnetic components of the electromagnetic field,E(r) andB(r) are given

by

E(r) =−∇φ(r), B(r) =∇×A(r).

It is assumed that, although the particles are charged, they do not interact

with each other, i.e. an ideal gas in an electromagnetic field. If the density is

low enough, this is not an unreasonable assumption, as the interaction with

the field will dominate over the Coulomb interaction.

a. Derive Hamilton’s equations for this system, and determine the force on

each particle in terms of the electric and magnetic fieldsE(r) andB(r),

respectively. This contribution to the force due to the field is knownas

theLorentz force. Express the equations of motion in Newtonian form.

b. SupposeN= 1, that the electric field is zero everywhere (E= 0), and

that the magnetic field is a constant in thezdirection,B= (0, 0 ,B). For

this case, solve the equations of motion for an arbitrary initial condition

and describe the motion that results.

1.14 Prove that the energy in eqn. (1.11.46) is conserved making useof eqn.

(1.11.47) for the torques.

∗1.15 (For amusement only): Consider a system with coordinateq, momentump,

and Hamiltonian

H=

pn

n

+

qn

n

,

wherenis an integer larger than 2. Show that if the energyEof the system

is chosen such thatnE=mn, wheremis a positive integer, then no phase

space trajectory can ever pass through a point for whichpandqare both

positive integers.