376 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRYthe constraints; i.e., the distribution 'D. One can introduce local coordinates qi =
(r^0 , sa) where a= 1, ... n - k, in which wa has the form
wa(q) = ds a + A~(r, s)dr^0 ,
where the summation convention is in force. Thus, we are locally writing the
distribution as
v = {(r, s,i', s) E TQ Is+ A~r^0 = O}.
The equations of motion, (3.1) may be rewritten by noting that the allowedvariations oqi = ( 8r^0 , 8sa) satisfy 8sa + A~8r^0 = 0. Substitution into (3.1) gives
(
d BL BL) _ Aa ( d BL BL)
dt Br^0 - Br^0 -^0 dt Bsa - Bsa.(3.2)
Equation (3.2) combined with the constraint equations
(3.3)
gives a complete description of the equations of motion of the system; this procedure
may be viewed as one way of eliminating the Lagrange multipliers. Using thisnotation, one finds that .A = AaWa, where
.A = !!: BL BL
a dt Bsa Bsa.
Equations (3.2) can be written in the following way:where!!:_ BLc - BLc + Aa BLc = - BL Bb rf3
dt Br^0 Br^0 0 Bsa Bsb o:f3 'L (r^0 Sa r^0 ) = L(r°' Sa r^0 -A a (r s)r^0 )
c ' ' ' ' ' 0: '(3.4)
is the coordinate expression of the constrained Lagrangian defined by L e = LIV
and whereLetting dwb be the exterior derivative of wb, a computation shows thatdwb ( q,.) = B~f3r^0 drf3
and hence the equations of motion have the form- 8L = (!!:... BLc - BLc A a BLc) 8 0: = - BL dwb(. 8 )
c dt Br 0 Br^0 +^0 Bsa r Bsb q, r.
(3.5)
This form of the equations isolates the effects of the constraints, and shows, in
particular, that in· the case where the constraints are integrable (i.e., d w = 0), the
equations of motion are obtained by substituting the constraints into the Lagrangian
and then setting the variation of L e to zero. However in the non-integrable case the
constraints generate extra (curvature) terms, which must be taken into account.
The above coordinate results can be put into an interesting and useful intrinsic
geometric framework. The intrinsically given information is the distribution and
the Lagrangian. Assume temporarily that there is a bundle structure 1fQ,R : Q ---+ R