derivative is zero: pk = 0. Therefore: The momentum conjugate to a cyclic

™

coordinate is conserved, i.e., it is a constant of the motion.

An example, worked out in detail in Section 15.6, is the center-of-mass

motion of a system of mutually interacting particles isolated in space: since

the Lagrangian cannot depend on the position of the center of mass, its

coordinates are cyclic, and the conjugate momentum, which is the total

linear momentum of the system, is conserved. Hence the center of mass can

only move with constant velocity.

It is not always true that the cyclic coordinate itself is moving with con-

stant velocity. An example is the motion of a diatomic molecule in free

space, where the rotation angle (in a plane) is a cyclic coordinate. The

conjugate momentum is the angular velocity multiplied by the moment of

2 We use the notation K for the kinetic energy rather than the usual T in order to avoid confusion

with the temperature T .

15.5 Coordinate transformations 401

inertia of the molecule. So, if the bond distance changes, e.g., by vibration,

the moment of inertia changes, and the angular velocity changes as well.

A coordinate that is constrained to a constant value, by some property of

the system itself or by an external action, also acts as a cyclic coordinate,

because it is not really a variable any more. However, its time derivative is

also zero, and such a coordinate vanishes from the Lagrangian altogether.

In Section 15.8 the equations of motion for a system with constraints will

be considered in detail.

15.5 Coordinate transformations

Consider a transformation from cartesian coordinates r to general coordi-

nates q:

r i = r i (q1 , . . . , qn ), i = 1, . . . , N, n = 3N. (15.13)

The kinetic energy can be written in terms of q:

N n N

1 1 ‚r i ‚r i

·

mi r 2 =

™

K= mi qk ql .

™™ (15.14)

2 2 ‚qk ‚ql

i=1 k,l=1 i=1

This is a quadratic form that can be expressed in matrix notation:3

K(q, q) = 1 qT M(q)q,

2™ ™

™ (15.15)

where

N

‚r i ‚r i

·

Mkl = mi . (15.16)

‚qk ‚ql

i=1

The tensor M, de¬ned in (15.16), is called the mass tensor or sometimes

the mass-metric tensor.4 The matrix M(q) is in general a function of the

coordinates; it is symmetric and invertible (det M = 0). Its eigenvalues are

the masses mi , each three-fold degenerate.

Now we consider a conservative system

L(q, q) = K(q, q) ’ V (q).

™ ™ (15.17)

3 We use roman bold type for matrices, a vector being represented by a column matrix, in

contrast to italic bold type for vectors. For example: v · w = vT w. The superscript T denotes

the transpose of the matrix.

The latter name refers to the analogy with the metric tensor gkl = i [(‚ri /‚qk ) · (‚ri /‚ql )]

4

which de¬nes the metric of the generalized coordinate system: the distance ds between q and

q + dq is given by (ds)2 = kl gkl dqk dql .

402 Lagrangian and Hamiltonian mechanics

The conjugate momenta are de¬ned by

‚K(q, q)

™

pk = = Mkl ql

™ (15.18)

‚qk

l

or

™

p = Mq, (15.19)

and the Lagrangian equations of motion are

‚L 1 ‚M ‚V

= qT q’

™ ™

pk =

™ . (15.20)

‚qk 2 ‚qk ‚qk

¨

By inserting (15.20) into (15.19) a matrix equation is obtained for q :

1 ‚M±β

‚V ‚Mk±

Mkl ql = ’ ’

¨ + q± q β ,

™™ (15.21)

‚qk 2 ‚qk ‚qβ

l ±,β

which has the general form

™

M¨ = T(q) + C(q, q), (15.22)

q

where T is a generalized force or torque, and C is a velocity-dependent

force that comprises the Coriolis and centrifugal forces. Apart from the

fact that these forces are hard to evaluate, we are confronted with a set of

equations that require a complexity of order n3 to solve. Recently more

e¬cient order-n algorithms have been devised as a result of developments

in robotics.

By inverting (15.19) to q = M’1 p, the kinetic energy can be written in

™

terms of p (using the symmetry of M):

K = 1 (M’1 p)T M(M’1 p) = 1 pT M’1 p, (15.23)

2 2

and the Hamiltonian becomes

H = pT q ’ L = pT M’1 p ’ K + V = 1 pT M’1 (q)p + V (q),

™ (15.24)

2

with the Hamiltonian equations of motion

‚H

= (M’1 p)k ,

qk =

™ (15.25)

‚pk

1 T ‚M’1

‚H ‚V

pk = ’ =’ p p’

™ . (15.26)

‚qk 2 ‚qk ‚qk

(Parenthetically we note that the equivalence of the kinetic energy in (15.20)

and (15.26) implies that

‚M’1 ‚M ’1

= ’M’1 M, (15.27)

‚qk ‚qk

15.6 Translation and rotation 403

which also follows immediately from ‚MM’1 /‚qk = 0. We will use this

relation in ano ther context.)

The term ’‚V /‚qk is a direct transformation of the cartesian forces F i =

’‚V /‚r i :

‚V ‚r i

’ Fi ·

= , (15.28)

‚qk ‚qk

i