This paper develops a way to bring shock waves—sharp, moving jumps in density, velocity and other fields—into Hamilton’s principle. Hamilton’s principle is a variational rule that underlies much of fluid mechanics, but its standard form assumes the fluid fields are smooth in space and time. The authors modify that principle so it can handle discontinuities along a moving interface, denoted Γ(t), separating two smooth flow regions.

For the barotropic Euler equations (a model in which pressure depends only on density), the authors add terms to the action that live only on the shock surface. They allow independent variations of the fields on each side of the shock and of the shock geometry itself. From the stationarity of this modified action they recover the Rankine–Hugoniot jump conditions for mass and momentum. The extra interface term can be interpreted as a dissipation potential associated with energy loss at shocks. The paper shows that this extra potential leads to a modified energy balance; in particular the potential is not generally just the volume of the domains, and in one-dimensional barotropic examples one cannot take a simple choice that would remove the modification.

The authors then extend the idea to the full compressible Euler equations, which include the specific entropy as a dynamic variable. They use a variational formulation inspired by nonequilibrium thermodynamics and impose advection-type constraints on an entropy-related variable (denoted ς). With these ingredients they obtain the Rankine–Hugoniot relations for mass, momentum and energy, and also equations involving entropy. In this full model the total energy satisfies an exact conservation law. That contrast—modified energy balance for barotropic models versus exact energy conservation for the full compressible model—is a central structural finding of the work.

Why this matters: the Rankine–Hugoniot conditions are the basic compatibility rules that shocks must satisfy. Deriving them from a single variational principle ties shock dynamics to the geometric and energetic structure behind Hamilton’s principle. This can clarify the meaning of dissipation at shocks and gives a foundation for future work on structure-preserving numerical methods, variational approximations of discontinuous solutions, and further theoretical analysis.