1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 5. VARIATIONAL INTEGRATORS 411

for a (local) section <P of Y. The left hand side of (5.46) is often denoted 8L/8¢A

and is called the Euler-Lagrange derivative of L.

The following assertions regarding a section <P of the bundle nxy : Y ----+ X are

then equivalent:

l. ¢ is a stationary point of J x .C( (j^1 ( ¢));

2. the Euler-Lagrange equations (5.46) hold in coordinates;


3. for any vector field W on J1 Y ,

(j1(¢))* (iwDi::) = 0. (5.47)

Example: A nonlinear wave equation. We consider a simple example to

illustrate what the multisymplectic formalism is about. Consider the nonlinear

wave equation

~:t -6¢ - N'(<P) = 0,

where 6 is the Laplace operator and N is a real-valued c= function of one variable.

Specialize now to one spatial dimension (the Laplace operator is then just the second
spatial partial derivative operator) so that n = 1, X = JR^2 , and the fibers of Y are
R The Lagrangian density for this equation is

In coordinates (x, t, ¢,<Pt, <Px) on J1(Y), the multisymplectic 3-form is:

D.c -d¢ A d<Pt A dx - d¢ A d¢x A dt

-N'(¢)d¢ A dx A dt

+<Pt d<Pt A dx A dt - <Px d<Px A dx A dt.

Sections of Y are mappings (¢(x, t) of JR^2 into JR, and sections of J(Y) are

mappings from JR^2 to JR^3. Concretely, the first jet of a section <P is j^1 ( ¢) ( x, t) : =

(<P(x, t), <Pt(x, t), <Px(x, t)). Write the conjugate momenta as p^1 = <Px and p^0 =<Pt·

The equation can be reformulated as

(5.48)

To each coefficient matrix J μ, we associate the contact form wμ on JR^3 given by

wμ(u 1 ,u 2 ) = (J μu 1 ,u 2 ), where u 1 ,u2 E JR^3 and(·,·) is the standard inner product

on JR^3. Thus, one sees in this example, the origin of the term "multisymplectic."
Gotay, Isenberg and Marsden [1998] and Marsden, Patrick and Shkoller [1998]
investigate the general theory of multisymplectic systems. One such interesting
question is the PDE version of symplecticity of the fl.ow, which is a general identity
that the linearized equations must satisfy. In this case, this reads:

:t [w^0 (J1(<Pt),j1(¢x))] + :x [w^1 (j1(<Pt),j^1 (¢x))] = 0.

Many basic equations in continuum mechanics (wave equations, shallow water equa-
tions, etc.) have been put into the multisymplectic formalism (see Marsden and
Shkoller [1997] for some simple examples and for links with the important work of
Bridges [1994, 1997]). Marsden, Patrick and Shkoller [1998] show how to obtain all