1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 5. VARIATIONAL INTEGRATORS 407

Given a solution q(t) E CL, a first variation at q(t) is a vector field V on

Q along q(t) that is the derivative of some curve in CL at q(t). When CL is a

manifold, a first variation is just a vector at q(t) tangent to CL. Temporarily define

a = d6 - 8 L where by abuse of notation 8 L is the one form on C defined by

8L(q(t))8q(t) = 8L(b)8q(b) - 8L(a)8q(a).

Then CL is defined by a = 0 and we have t he equation

d6=a+8L,

so if V and Ware first variations at q(t), we obtain

0 = V _J W _J d^2 6 = V _J W _J d a + V _J W _J d 8 L.

We have the identity

da(V, W) (q(t)) = V(a(W)) - W(a(V)) - a([V, W]),

(5.25)

(5.26)

which we will use to evaluate (5.25) at the curve = q(t). Let qy (t) be a family of

solutions tangent to V at E = 0. For t he first term of (5.26), we have

which vanishes, since a is zero along qy for every E. Similarly the second term

of (5.25) at q(t) also vanishes, while the third term of vanishes since a (q(t)) = 0.

Consequently, symplecticity of t he the Lagrangian flow Ft may be written

V _J W _J d8 L = 0,

for all first variations V and W. This formuation is valid whether or not the

solution space is a manifold, and it does not explicitly refer to any temporal notion.

Similarly, Noether 's theorem may b e written in this way. Summarizing, using t he

variational principle, the an alogue that the evolution is symplectic is the equation

d^26 = 0 restricted to first variations of the space of solut ions of the variational

principle. The analogue of Noether's theorem is infinitesimal invariance of d6

restricted to first variations of t he space of solutions of t he variational principle.
The variational route to the differential-geometric formalism has obvious ped-
agogical advantages. More than that, however, it systematizes searching for the
corresponding formalism in other contexts. In fact, Marsden, Patrick and Shkoller
[1998] show how this works in the context of classical field theory and multisymplec-
tic geometry; i .e., for Lagrangian and Hamiltonian pde's such as nonlinear wave
equations.

Veselov discretizations of mechanics. We now show how the discrete La-
grangian formalism in Veselov [1988], [1991] and Moser and Veselov [1991] described
earlier fits into this variational framework. Recall that a discrete Lagrangian is
a smooth map lL : Q x Q --+IR, and the corresponding action is
n-1
§ = 2: JL(qk+l, qk)·
k=O

The discrete variational principle is to extremize 6 for variations holding the
endpoints q 0 and qn fixed. This variational principle determines a "discrete flow"