  The equations of Lagrangian, ideal, one-dimensional (1D), compressible gas
dynamics are written in a multi-symplectic form using the Lagrangian mass
coordinate $m$ and time $t$ as independent variables, and in which the Eulerian
position of the fluid element $x=x(m,t)$ is one of the dependent variables.
This approach differs from the Eulerian, multi-symplectic approach using
Clebsch variables. Lagrangian constraints are used to specify equations for
$x_m$, $x_t$ and $S_t$ consistent with the Lagrangian map, where $S$ is the
entropy of the gas.
  We require $S_t=0$ corresponding to advection of the entropy $S$ with the
flow. We show that the Lagrangian Hamiltonian equations are related to the de
Donder-Weyl multi-momentum formulation. The pullback conservation laws and the
symplecticity conservation laws are discussed. The pullback conservation laws
correspond to invariance of the action with respect to translations in time
(energy conservation) and translations in $m$ in Noether's theorem. The
conservation law due to $m$-translation invariance gives rise to a novel
nonlocal conservation law involving the Clebsch variable $r$ used to impose
$\partial S(m,t)/\partial t=0$.
  Translation invariance with respect to $x$ in Noether's theorem is associated
with momentum conservation. We obtain the Cartan-Poincar\'e form for the
system, and use it to obtain a closed ideal of two-forms representing the
equation system.
