Physical Chemistry Third Edition

E Classical Mechanics 1271

The second term in Eq. (E-15) produces a rate of change in

.

ρifV does not depend

onρ. This represents thecentrifugal force, which is not a force, but an expression of

the natural tendency of an orbiting particle to move off in a straight line. To maintain

a circular orbit about the origin of coordinates, the second term must be canceled by a

centripetal force:

F(centripetal)−

∂V

∂ρ

−mρ

.

φ

2

−

mv^2

ρ

(E-17)

where the speedvin a circular orbit equalsρ

.

φ.

An important variable for an orbiting object is theangular momentumaround a fixed

center. This is the vector

Lmr×v (E-18)

whereris the position vector from the fixed center to the particle,vis its velocity

vector, and×stands for the cross product of two vectors, discussed in Appendix B.

Figure E.2 illustrates the angular momentum of a single orbiting particle. In a circular

orbit,randvare perpendicular to each other and the angular momentum vector has

the magnitude

L|L|mρvmρ^2

.

φ (E-19)

For any moving mass not subject to friction or forces other than a centripetal force,

the angular momentum about a fixed center remains constant. We say that it iscon-

served. The orbit of the moving mass remains in the same plane, and the angular

momentum has a fixed direction and a fixed magnitude. If a system consists of sev-

eral interacting particles, the vector sum of their angular momenta (the total angular

momentum) is conserved. If the set of particles constitutes a rotating rigid body such

as a gyroscope spinning on its axis, the total angular momentum remains constant if no

forces act on the body. Figure E.3 shows a simple gyroscope. If a gyroscope stands on

one end of its axis at an angle in a gravitational field, the gravitational force produces a

torque (a turning force) in the direction that would make the gyroscope fall on its side

if it were not rotating. If it is rotating, instead of making the gyroscope fall on its side,

it makes the axis move (precess) around a vertical cone, as shown in the figure.

Cone of directions

ofL as L precesses

in a gravitational field

L

Supporting object

Figure E.3 A Simple Gyroscope.If

a torque is placed on a gyroscope,

as by supporting one end of the gyro-

scope in a gravitational field, the gyro-

scope will precess, which means that

its axis of rotation will move around a

cone with a vertical axis.

E.4 Hamiltonian Mechanics

Hamilton’s method is similar to that of Lagrange in that it provides equations of motion

that have the same form in any coordinate system. It usesconjugate momentainstead

of time derivatives of coordinates as state variables. The conjugate momentum to the

coordinateqiis defined by

pi

∂L

∂

.

qi

(i1, 2,...,n) (definition) (E-20)

whereLis the Lagrangian. The momentum conjugate to a Cartesian coordinate is a

component of the ordinary (linear) momentum:

pxmvx, pymvy, pzmvz (E-21)