This paper shows how to extend Hamilton’s principle — a basic variational rule used to derive fluid equations — so it can handle shock waves. Classical Hamilton’s principle assumes the flow is smooth. That assumption breaks down at shocks, which are moving surfaces where density, velocity, and other quantities jump abruptly. The authors build a version of the action principle that directly accounts for those jumps.

For the barotropic compressible Euler equations (models where pressure depends only on density), the authors add terms to the action that live on the shock surface. They treat the shock surface Γ(t) as an evolving interface that separates two smooth regions and allow independent variations on each side. From stationarity of the modified action they recover the Rankine–Hugoniot jump conditions for mass and momentum. The extra interface term can be read as a dissipation potential: it captures the energy loss associated with shocks and leads to a modified energy balance that includes an interface contribution V_shock.

For the full compressible Euler system (which includes entropy and internal energy), the authors use a variational formulation of nonequilibrium thermodynamics together with suitable variational and phenomenological constraints. In that setting they recover the full Rankine–Hugoniot relations for mass, momentum, and energy, and the total energy satisfies exact conservation. The paper highlights a structural difference: barotropic models need an added interface potential to account for energy change, while the full model’s entropy degree of freedom lets general interface variations be accommodated without modifying total energy.

Technically, the work is geometric. The shock surface Γ(t) and the two regions it separates, denoted M±(t), are part of the configuration. Variations of the bulk fields and of the interface geometry are allowed. The authors work out the resulting bulk equations, the jump conditions across Γ(t), and boundary conditions as critical-point conditions of the action. They illustrate aspects of the approach in one-dimensional flows and outline extensions that include heat conduction.