  We introduce a variational time discretization for the multi-dimensional gas
dynamics equations, in the spirit of minimizing movements for curves of maximal
slope. Each timestep requires the minimization of a functional measuring the
acceleration of fluid elements, over the cone of monotone transport maps. We
prove convergence to measure-valued solutions for the pressureless gas dynamics
and the compressible Euler equations. For one space dimension, we obtain sticky
particle solutions for the pressureless case.
