Summary: We have developed a novel high-order, energy conserving approach for solving the Euler equations in a moving Lagrangian frame, which is derived from a general finite element framework. Traditionally, such equations have been solved by using continuous linear representations for kinematic variables and discontinuous constant fields for thermodynamic variables; this is the so-called staggered grid hydro (SGH) method. From our general finite element framework, we can derive several specific high-order discretization methods and in this paper we introduce a curvilinear finite element method which uses continuous bi-quadratic polynomial bases (the \(Q_{2}\) isoparametric elements) to represent the kinematic variables combined with discontinuous (mapped) bi-linear bases to represent the thermodynamic variables. We consider this a natural generalization of the SGH approach and show that under simplifying low-order assumptions, we exactly recover the classical SGH method. We review the key parts of the discretization framework and demonstrate several practical advantages to using curvilinear finite elements for Lagrangian shock hydrodynamics, including: the ability to more accurately capture geometrical features of a flow region, significant improvements in symmetry preservation for radial flows, sharper resolution of a shock front for a given mesh resolution including the ability to represent a shock within a single zone and a substantial reduction in mesh imprinting for shock waves that are not aligned with the computational mesh.