From Stationary Action to Least Constraint: A Higher-Order Variational Bridge in Classical Mechanics

Classical mechanics has two commonly cited variational principles: the principle of stationary action and Gauss's principle of least constraint. Here we show that these principles arise as two limits of a single higher-order variational framework based on a mass-weighted residual action. When applied like a "boundary-value problem," the framework recovers the standard Euler–Lagrange and Hamiltonian equations of motion that follow from the stationary-action principle. When treated instead like an "initial value problem" with a minimization with fixed position and velocity, it reduces to Gauss's principle, yielding the constrained acceleration that minimizes the mass-weighted deviation from free motion. This unified formulation reveals that action and constraint principles share a common variational origin, differing only in their temporal "boundary conditions." The result offers new perspectives for variational integrators and constrained dynamics.