and is therefore a kind of generalized force on qk . Note, however, that

in general the time derivative of pk is not equal to this generalized force!

Equality is only valid in the case that the mass tensor is independent of qk :

‚M ‚V ‚r i

= 0 then pk = ’ Fi ·

if ™ = . (15.29)

‚qk ‚qk ‚qk

i

15.6 Translation and rotation

Consider a system of N particles that interact mutually under a poten-

tial V int (r 1 , . . . , r N ), and are in addition subjected to an external potential

V ext (r 1 , . . . , r N ). Homogeneity and isotropy of space dictate that neither

the kinetic energy K nor the internal potential V int can depend on the over-

all position and orientation of the system of particles. As we shall see, these

properties give a special meaning to the six generalized coordinates of the

overall position and orientation. Their motion is determined exclusively by

the external potential. In the absence of an external potential these coordi-

nates are cyclic, and their conjugate moments “ which are the total linear

and angular momentum “ will be conserved. It also follows that a gen-

eral three-dimensional N -body system has no more than 3N ’ 6 internal

coordinates.5

15.6.1 Translation

Consider6 a transformation from r to a coordinate system in which q1 is a

displacement of all coordinates in the direction speci¬ed by a unit vector

n : dr i = n dq1 . Hence, for any i,

™

‚r i ‚ ri

= n, = n. (15.30)

‚q1 ‚ q1

™

5 For a system consisting of particles on a straight line, as a diatomic molecule, one of the

rotational coordinates does not exist and so there will be at most 3N ’ 5 internal coordinates.

6 We follow the line of reasoning by Goldstein (1980).

404 Lagrangian and Hamiltonian mechanics

Homogeneity of space implies that

‚V int

‚K(q, q)

™

= 0, = 0. (15.31)

‚q1 ‚q1

For the momentum p1 conjugate to q1 follows

‚ r2

™ ™

‚K 1 ‚ ri

mi r i · =n·

mi i = ™

p1 = = mi v i , (15.32)

‚ q1

™ 2 ‚ q1

™ ‚ q1

™

i i i

which is the component of the total linear momentum in the direction n.

Its equation of motion is

‚V ext ‚V ext ‚r i

p1 = ’ =’ · · n.

F ext

™ = (15.33)

i

‚q1 ‚r i ‚q1

i i

So the motion is governed by the total external force. In the absence of an

external force, the total linear momentum will be conserved.

The direction of n is immaterial: therefore there are three independent

general coordinates of this type. These are the components of the center of

mass (c.o.m.)

1

def

r cm = mi r i , with M = mi , (15.34)

M

i i

with equation of motion

1 1

F ext .

™

r cm = mi v i = (15.35)

M M

i i

In other words: the c.o.m. behaves like a single particle with mass M .

15.6.2 Rotation

Consider a transformation from r to a coordinate system in which dq1 is a

rotation of the whole body over an in¬nitesimal angle around an axis in the

direction speci¬ed by a unit vector n : dr i = n — r 1 dq1 . Hence, for any i

™

‚r i ‚ ri

= n — ri .

= (15.36)

‚q1 ‚ q1

™

Isotropy of space implies that

‚V int

‚K(q, q)

™

= 0, = 0. (15.37)

‚q1 ‚q1

15.7 Rigid body motion 405

For the momentum p1 conjugate to q1 it follows that7

‚K 2

mi ‚ r1

™i

· (n — r i )

1

p1 = = = i mi v i

i

2 ‚q ™

‚ q1

™

mi (r i — v i ) · n = L · n,

= (15.38)

i

where the angular momentum of the system. is given by

def

mi r i — v i .

L= (15.39)

i

In general, L depends on the choice of the origin of the coordinate system,

except when the total linear momentum is zero.

The equation of motion of p1 is

‚V ext

p1 = ’ F ext · (n — r i ) = r i — F ext · n = T · n, (15.40)

™ = i i

‚q1

i i

where the torque exerted on the system is given by

def

r i — F ext .

T= (15.41)

i

i