SERGEY PEKARSKY
Control and Dynamical Systems
Mail Code 107-81
California Institute of Technology
Pasadena, CA 91125, USA
http://www.cds.caltech.edu/~sergey
sergey@cds.caltech.edu
Abstract:
The hydrodynamics of ideal fluids is put into the multisymplectic framework.
The configuration space is a bundle $Y$ over the
space-time manifold $X := M \times \mathbb{R}$ with fibers diffeomorphic to
the ``fluid container'' $M$. The system is defined by a constrained
Lagrangian on a first jet bundle $J^1 Y$, with the constraint corresponding to
the incompressibility condition. A section $y$ of $Y$ that is a
solution of Euler-Lagrange equations can be thought of as a surface over $X$
which captures the volume preserving diffeomorphisms $\eta \in
\mathcal{D}_\mu (M)$ for all $(x,t) \in X$ and upon space-time split
corresponds to the fluid flow.
The ``particle relabeling'' symmetry, which is intrinsic to fluid dynamics, can be realized as a purely horizontal symmetry. The corresponding multi-momentum map and its Noether current are computed. Moreover, the Noether's conservation theorem is shown to be equivalent to Euler-Lagrange equations.