Reduction of symmetric time-dependent Hamiltonian systems I: presymplectic principal $\mathbb{R}$-bundles

What the math gives to ML

The paper gives a concrete quotient construction for time-dependent Hamiltonian mechanics: the extended cotangent bundle contains a redundant affine momentum coordinate conjugate to time, and quotienting by its principal R-action produces the physical vertical cotangent state. The transferable asset is an explicit gauge-like representation in which Hamiltonian sections select one representative from each affine fiber, preventing a model from fitting an unidentifiable energy coordinate. This suggests Hamiltonian neural networks that predict only reduced momenta and reconstruct an extended covector through a learned section. The construction also supports symmetry losses for transformations that translate time through a group character or Lie-algebra cocycle.