1549380323-Statistical Mechanics Theory and Molecular Simulation

10 Classical mechanics

1.4 Lagrangian formulation of classical mechanics: A general

framework for Newton’s laws

Statistical mechanics is concerned with characterizing the numberof microscopic states
available to a system and, therefore, requires a formulation of classical mechanics
that is more closely connected to the phase space description thanthe Newtonian
formulation. Since phase space provides a geometric description ofa system in terms of
positions and momenta, or equivalently in terms of positions and velocities, it is natural
to look for an algebraic description of a system in terms of these variables. In particular,
we seek a “generator” of the classical equations of motion that takes the positions and
velocities or positions and momenta as its inputs and produces, through some formal
procedure, the classical equations of motion. The formal structure we seek is embodied
in theLagrangianandHamiltonianformulations of classical mechanics (Goldstein,
1980). The introduction of such a formal structure places some restrictions on the form
of the force laws. Specifically, the forces are required to beconservative. Conservative
forces are defined to be vector quantities that are derivable froma scalar function
U(r 1 ,...,rN), known as apotential energy function, via

Fi(r 1 ,...,rN) =−∇iU(r 1 ,...,rN), (1.4.1)

where∇i=∂/∂ri. Consider the work done by the forceFiin moving particleifrom
points A to B along a particular path. This work is

WAB=

∫B

A

Fi·dl. (1.4.2)

SinceFi=−∇iUis conserved, the line integral simply becomes the difference in
potential energy between path endpoints A and B,WAB=UA−UB, independent of
the path taken. Because the work is independent of the path, it follows that along a
closed path ∮

Fi·dl= 0. (1.4.3)

Given theNparticle velocities,r ̇ 1 ,...,r ̇N, the kinetic energy of the system is given
by

K(r ̇ 1 ,...,r ̇N) =

1

2

∑N

i=1

mir ̇^2 i. (1.4.4)

TheLagrangianLof a system is defined as the difference between the kinetic and
potential energies expressed as a function of positions and velocities:

L(r 1 ,...,rN,r ̇ 1 ,...,r ̇N) =K(r ̇ 1 ,...,r ̇N)−U(r 1 ,...,rN). (1.4.5)

The Lagrangian serves as the generator of the equations of motion via theEuler–
Lagrangeequation:
d
dt

(

∂L

∂r ̇i

)

−

∂L

∂ri

= 0. (1.4.6)