Dirac structures on tangent bundles: a geometric framework for variational principles, constrained dynamics, and symmetry reduction

We introduce a Dirac structure on the tangent bundle of a configuration manifold, called a $\mathit{\text{Lagrange–Dirac}}\mathit{\text{structure}}$, which is naturally induced by the Lagrangian two-form associated with a (possibly degenerate) Lagrangian and a constraint distribution. This structure provides a unified geometric framework for Lagrange–Dirac dynamical systems, encompassing nonholonomic, degenerate Lagrangian, and symmetric systems. In the hyperregular case, the system recovers a first-order formulation of the Lagrange–d’Alembert equations. Although nonholonomic dynamics does not preserve the Lagrangian two-form, we show that the underlying Lagrange–Dirac structure is preserved up to gauge transformations, yielding a natural gauge covariance property. We also formulate an intrinsic variational principle directly on the tangent bundle, referred to as the $\mathit{\text{Lagrange–d’Alembert–Dirac}}\mathit{\text{principle}}$, which recovers Hamilton’s principle in the unconstrained case and the Lagrange–d’Alembert principle in the hyperregular constrained case. Furthermore, we develop a reduction theory for systems with Lie group symmetry, deriving a reduced Lagrange–Dirac structure over the Lie algebra that yields the Euler–Poincaré–Dirac equations and a corresponding reduced variational principle. Finally, we illustrate the theory through examples including charged particles, electric circuits, and systems with Lagrangians linear in velocity, and present an infinite-dimensional extension to ideal fluids that naturally incorporates the incompressibility constraint and recovers the Euler equations.