Phase volume and canonicity preservation in finite-difference gas dynamic schemes constructed with the sequential variational method

The paper proves that the phase volume and canonicity (Hamiltonicity) are preserved in the finite-difference schemes of Lagrangiam gas dynamics constructed using the sequential variational method with Hamilton–Ostrogradskii principle of least action, which is discrete in time and space. It has been proved that in implicit finite-difference schemes with permanent “weight” $\theta=1/2$ (in the equations for coordinates and velocity) the phase volume and canonicity are not preserved for an arbitrary varying time step independently of the way of selecting “hidden” generalized coordinates and “hidden” generalized momenta (such difference schemes cannot be constructed using the sequential variational method).