Physical Chemistry , 1st ed.

The utility of the Hamiltonian function depends on the kinetic energy K,
which is a function of the time derivatives of position, that is, the velocities. If
Kwere to depend on the sum of the squareof velocities:

K

N

j 1

cjq ̇j^2 (9.12)

(where the cjvalues are the expansion coefficients of the individual compo-
nents ofK) then it can be shown that the Hamiltonian function is

HKV (9.13)

That is,His simply the sum of the kinetic and potential energies. The kinetic
energy expressions that we consider here are indeed of the form in equation
9.12. The Hamiltonian function conveniently gives the total energy of the sys-
tem,a quantity of fundamental importance to scientists. The Hamiltonian
function can be differentiated and separated to show that





H

pj

q ̇j (9.14)





H

qj

p ̇j (9.15)

These last two equations are Hamilton’s equations of motion. There are two
equations for each of the three spatial dimensions. For one particle in three di-
mensions, equations 9.14 and 9.15 give six first-order differential equations
that need to be solved in order to understand the behavior of the particle. Both
Newton’s equations and Lagrange’s equations require the solution of three
second-order differential equations for each particle, so that the amount of cal-
culus required to understand the system is the same. The only difference lies
in what information one knows to model the system or what information one
wants to get about the system. This determines which set of equations to use.
Otherwise, they are all mathematically equivalent.

Example 9.1

Show that, for a simple one-dimensional Hooke’s-law harmonic oscillator

having mass m, the three equations of motion yield the same results.

Solution

For a Hooke’s-law harmonic oscillator, the (nonvector) force is given by

Fkx

and the potential energy is given by

V^12 kx^2

where kis the force constant.

a.From Newton’s laws, a body in motion must obey the equation

Fmd

dt

(^2) x
 2
and the two expressions for force can be equated to give
md
dt
(^2) x
 2 kx
9.2 Laws of Motion 245