A new mathematical preprint sets out a way to simplify time-dependent Hamiltonian systems with symmetries while retaining the geometric link between their extended and restricted phase spaces. The procedure combines cotangent-bundle reduction with the reduction of presymplectic structures. Here, corank refers to the number of directions in the structure's kernel, or degenerate part: two in the extended setting and one in the restricted setting.
The work treats the extended momentum space as a principal real-line bundle over the restricted momentum space. Put simply, the construction keeps the two spaces connected by a bundle structure rather than treating them as separate objects. The paper shows that this relationship descends to the reduced spaces as well.
A route for symmetries that move the clock
The approach is designed for symmetries that can translate time. It links invariance of the time one-form, the geometric object used to encode time in the construction, to a multiplicative function on the symmetry group and an associated Lie-algebra one-cocycle, an algebraic consistency condition for the symmetry.
The paper constructs momentum maps on both phase spaces. These are the geometric devices used to express how a system's symmetry appears in its phase-space variables. In the extended space, the map includes a cocycle correction to the usual cotangent momentum map. In the restricted space, the map is induced through the bundle projection from the extended construction.
The framework also gives a time-dependent version of a Noether first integral. For each element of the system's symmetry algebra, the corresponding component of the restricted momentum map is conserved along the time-dependent dynamics, even though its formula may depend explicitly on time.
What remains after reduction
At a regular momentum value, the relevant level sets can be quotiented by the symmetry to produce reduced presymplectic spaces. The reduced extended space has corank two, while the reduced restricted space has corank one. These are the same expected coranks carried by the unreduced spaces in the paper's construction.
The reduced kernels, the directions left degenerate by the presymplectic forms, are not left without structure. In the extended space, the reduced kernel is generated by the reduced principal-action field together with the reduced Hamiltonian field. In the restricted space, it is generated by the reduced time-dependent field.
The reduced extended space still maps to the reduced restricted space as a principal real-line bundle. The Hamiltonian section also descends to this reduced bundle, and the reduced presymplectic forms remain linked through that section. The result preserves the main geometric relationship while replacing the original system with its symmetry-reduced counterpart.
Examples and boundaries
The paper illustrates the construction with a time-dependent N-dimensional harmonic oscillator and a time-dependent Elroy's Beanie. These worked mechanical examples show how the formal reduction framework is applied in specific systems.
The statements are conditional on the paper's assumptions. The momentum-map results are given under equivariance assumptions, while the reduced spaces are constructed at a regular momentum value. The paper also excludes relative equilibria of the homogeneous Hamiltonian vector field when asserting the expected coranks. Their structure and stability are left for future work.
The document is an arXiv preprint, version 3, dated 14 September 2026. Its stated corank results do not cover the relative equilibria excluded by the paper's assumptions.
Paper data and sources
Original title: Reduction of symmetric time-dependent Hamiltonian systems I: presymplectic principal $\mathbb{R}$-bundles
Authors: C. Benítez, D. Iglesias Ponte, J. C. Marrero, E. Padrón
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text