1549380323-Statistical Mechanics Theory and Molecular Simulation

50 Classical mechanics

1.13 Problems

1.1. Solve the equations of motion given arbitrary initial conditions for a one-

dimensional particle moving in a linear potentialU(x) =Cx, whereCis a

constant, and sketch a representative phase space plot.

∗1.2. A particle of unit mass moves in a potential of the form

U(x) =−

ω^2

8 a^2

(

x^2 −a^2

) 2

.

a. Show that the function

x(t) =atanh[(t−t 0 )ω/2]

is a solution to Hamilton’s equations for this system, wheret 0 is an arbi-

trary constant.

b. Let the origin of time bet=−∞rather thant= 0. To what initial

conditions does this solution correspond?

c. Determine the behavior of this solution ast→∞.

d. Sketch the phase space plot for this particular solution.

1.3. Determine the trajectoryr(t) for a particle of massmmoving in three di-

mensions subject to a central potential of the formU(r) =kr^2 /2. Verify your

solution for different values ofland given values ofmandkby numerically

integrating eqn. (1.4.32). Discuss the behavior of the solution for different

values ofl.

1.4. Repeat problem 3 for a potential of the formU(r) =κ/r.

∗1.5. Consider Newton’s equation of motion for a one-dimensional particle subject

to an arbitrary force,m ̈x=F(x). A numerical integration algorithm for the

equations of motion, known as the velocity Verlet algorithm (see Chapter 3),

for a discrete time step value ∆tis

x(∆t) =x(0) + ∆t

p(0)

m

+

∆t^2

2 m

F(x(0))

p(∆t) =p(0) +

∆t

2

[F(x(0)) +F(x(∆t))].

By considering the Jacobian matrix

J =







∂x(∆t)

∂x(0)

∂x(∆t)

∂p(0)

∂p(∆t)

∂x(0)

∂p(∆t)

∂p(0)





,

show that the algorithm is symplectic, and show that det[J] = 1.