1549380323-Statistical Mechanics Theory and Molecular Simulation

Lagrangian formulation 15

choosing a coordinate frame in which thezaxis lies along the direction ofl. In such a
frame, the motion occurs solely in thexyplane so thatθ=π/2 andθ ̇= 0. With this
simplification, the equations of motion become

m ̈r−mrφ ̇^2 =−

dU

dr

mr^2 φ ̈+ 2mrr ̇φ ̇= 0. (1.4.28)

The second equation can be expressed in the form

d

dt

(

1

2

r^2 φ ̇

)

= 0, (1.4.29)

which expresses another conservation law known as the conservation ofareal velocity,
defined as the area swept out by the radius vector per unit time. Setting the quantity,
mr^2 φ ̇=λ, whereλis constant, the first equation of motion can be written as

m ̈r−

λ^2

mr^3

=−

dU

dr

. (1.4.30)

Since the total energy

E=

1

2

m

(

r ̇^2 +r^2 φ ̇^2

)

+U(r) =

1

2

mr ̇^2 +

λ^2

2 mr^2

+U(r) (1.4.31)

is conserved, eqn. (1.4.31) can be inverted to give an integral expression

dt=

dr

√

2

m

(

E−U(r)− λ

2

2 mr^2

)

t=

∫r

r(0)

dr′

√

2

m

(

E−U(r′)− λ

2

2 mr′^2

), (1.4.32)

which, for certain choices of the potential, can be integrated analytically and inverted
to yield the trajectoryr(t).

1.4.2 Example: Two-particle system

Consider a two-particle system with massesm 1 andm 2 , positionsr 1 andr 2 , and
velocitiesr ̇ 1 andr ̇ 2 subject to a potentialUthat is a function of only the distance
|r 1 −r 2 |between them. Such would be the case, for example, in a diatomic molecule.
The Lagrangian for the system can be written as

L=

1

2

m 1 r ̇^21 +

1

2

m 2 r ̇^22 −U(|r 1 −r 2 |). (1.4.33)

Although such a system can easily be treated directly in terms of theCartesian posi-
tionsr 1 andr 2 , for which the equations of motion are

m 1 ̈r 1 =−U′(|r 1 −r 2 |)

r 1 −r 2

|r 1 −r 2 |