# Vorticity and Symplecticity in Multi-Symplectic Lagrangian Gas Dynamics

Research paper by **G. M. Webb, S. C. Anco**

Indexed on: **19 Jan '16**Published on: **19 Jan '16**Published in: **Mathematical Physics**

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#### Abstract

The Lagrangian, multi-dimensional, ideal, compressible gasdynamic equations
are written in a multi-symplectic form, in which the Lagrangian fluid labels,
$m^i$ (the Lagrangian mass coordinates) and time $t$ are the independent
variables, and in which the Eulerian position of the fluid element ${\bf
x}={\bf x}({\bf m},t)$ and the entropy $S=S({\bf m},t)$ are the dependent
variables. Constraints in the variational principle are incorporated by means
of Lagrange multipliers. The constraints are: the entropy advection equation
$S_t=0$, the Lagrangian map equation ${\bf x}_t={\bf u}$ where ${\bf u}$ is the
fluid velocity, and the mass continuity equation which has the form $J=\tau$
where $J=\det(x_{ij})$ is the Jacobian of the Lagrangian map in which
$x_{ij}=\partial x^i/\partial m^j$ and $\tau=1/\rho$ is the specific volume of
the gas. The internal energy per unit volume of the gas
$\varepsilon=\varepsilon(\rho,S)$ corresponds to a non-barotropic gas. The
Lagrangian is used to define multi-momenta, and to develop de-Donder Weyl
Hamiltonian equations. The de Donder Weyl equations are cast in a
multi-symplectic form. The pullback conservation laws and the symplecticity
conservation laws are obtained. One class of symplecticity conservation laws
give rise to vorticity and potential vorticity type conservation laws, and
another class of symplecticity laws are related to derivatives of the
Lagrangian energy conservation law with respect to the Lagrangian mass
coordinates $m^i$. We show that the vorticity-symplecticity laws can be derived
by a Lie dragging method, and also by using Noether's second theorem and a
fluid relabelling symmetry which is a divergence symmetry of the action. We
obtain the Cartan-Poincar\'e form describing the equations and we discuss a set
of differential forms representing the equation system.