Weak KAM theorems for subriemannian Lagrangians depending on the unknown function

What the math gives to ML

The paper supplies a constructive convex-duality template for dynamics constrained to a horizontal distribution rather than the full tangent space. Its transferable asset is the combination of superlinear control cost, uniform strict convexity, and bounded strictly decreasing dependence on a scalar state variable, which produces a well-behaved contact Hamiltonian. This can be turned into a neural ODE or state-space layer whose latent motion is restricted to learned or prescribed vector fields and whose scalar energy or confidence variable induces controlled dissipation. The most direct test is a contact-Hamiltonian latent dynamics module against unconstrained neural ODEs and Hamiltonian neural networks on trajectory prediction and long-horizon classification.