1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

404 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY

for all oq(t) with oq(a) = oq(b) = o. Abbreviating q, = q+Eoq, and using integration

by parts, the calculation is

d<S(q(t)). oq(t)

d

d I 1b L (q!(t), ~q! (t)) dt

E c=O a t

1

b. ( fJL d fJL) fJL · 1b

a oq' fJqi - dt fJqi dt + fJqi oq' a. (5.19)

The last term in (5.19) vanishes since oq(a) = oq(b) = 0, so that the requirement

(5.18) for 6 to be stationary yields the Euler-Lagrange equations

fJL _ ~ fJL = O. (5.20)

aq' dt aiJ.'

Recall that L is called regular when the matrix [8^2 L/fJqifJqJ] is everywhere a
nonsingular matrix and in this case, the Euler-Lagrange equations are second order
ordinary differential equations.
Since the action (5.17) is independent of the choice of coordinates, the Euler-
Lagrange equations are coordinate independent as well. Consequently, it is natural
that the Euler-Lagrange equations may be intrinsically expressed using the language
of differential geometry. This intrinsic development of mechanics is now standard,
and can be seen, for example, in Arnold [1989], Abraham and Marsden [1978], and
Marsden and Ratiu [1998].
The canonical 1-form on the 2n-dimensional cotangent bundle T * Q is defined
by
Go(aq)Waq = O'.q · T7rQWaq'

where O'.q E r;Q, and Waq E TaqTQ, and 7rQ : TQ----+ Q is the projection. The

Lagrangian L intrinsically defines a fiber preserving bundle map lFL: TQ----+ T*Q,
the Legendre transformation, by vertical differentiation:

lFL(vq)wq = dd I L(vq + Ewq)·
E E=O
One normally defines the Lagrange 1-formon TQ by pull-back GL = lFL*Go, and
the Lagrange 2-form by DL = -dGL. We then seek a vector field Xe (called the

Lagrange vector field) on TQ such that X e _J DL = dE, where the energy Eis

defined by E(vq) = lFL(vq)Vq - L(vq)· A number of these definitions can be given a
little more directly on TQ, without resorting to T*Q, but it amounts to the same
thing.

If lFL is a local diffeomorphism then X e exists and is unique, and its integral

curves solve the Euler-Lagrange equations. In addition, the fl.ow Ft of X e is sym-

plectic; that is, preserves DL: FtDL = nL. These facts are usually proved using

differential forms and Lie derivatives.
Despite the compactness and precision of this differential-geometric approach,
it is difficult to motivate and also is not entirely contained on the Lagrangian side.
The canonical 1-form 80 seems to appear from nowhere, as does the Legendre
transform lF L.

The variational approach. Besides being more faithful to history, more and
more, one is finding that there are advantages to staying on the "Lagrangian side".
Many examples can be given, but the theory of Lagrangian reduction discussed in
lecture 1 (the Euler-Poincare equations being an instance) and associated control