Hamiltonian reduction from particular integrals

Technical Summary: Hamiltonian Reduction from Particular Integrals

Problem Statement
The paper addresses the geometric mechanisms underlying "particular integrals"—quantities that are not globally conserved but become conserved when the dynamics of a Hamiltonian system is restricted to a specific invariant subset of phase space. While classical Liouville–Arnold integrability relies on global constants of motion in involution to define invariant tori, and previous work on particular integrability focused on solving restricted equations via Lie quadratures (which do not necessarily yield Hamiltonian systems), there is a gap in understanding how particular integrals can generate a rigorous Hamiltonian reduction mechanism. Specifically, the authors seek to determine if systems of particular integrals can define invariant submanifolds that inherit a presymplectic structure, allowing for a projection to a lower-dimensional, genuinely Hamiltonian system.

Methodology
The authors develop a geometric framework based on the following steps:

1. Generalization of Particular Integrals: The standard scalar condition for a particular integral, $\dot{f} = af$, is generalized to a system of functions $f_1, \dots, f_k$ satisfying a linear closure condition:
   $$\dot{f}_i = a^j_i f_j$$
   where $a^j_i$ are smooth functions on the phase space. Lemma 1 proves that the common zero level set $M_f = \{x \mid f_i(x) = 0\}$ of such a system is a dynamically invariant embedded submanifold of codimension $k$.

2. Presymplectic Reduction: In the Hamiltonian context, if the functions $f_i$ are in Poisson involution ($\{f_i, f_j\} = 0$), the invariant submanifold $M_f$ inherits a closed 2-form of constant rank (a presymplectic form) from the ambient symplectic manifold. The dynamics restricted to $M_f$ are presymplectic.

3. Gauge Reduction and Projection: The authors apply the gauge reduction procedure to the presymplectic manifold. By quotienting $M_f$ by the characteristic distribution (the kernel of the presymplectic form), they construct a reduced symplectic manifold. The dynamics on this quotient are shown to be a genuine Hamiltonian system with fewer degrees of freedom.

4. Lift Constructions: The paper investigates "lifts" of natural Hamiltonian systems (systems with kinetic and potential energy) to extended configuration spaces. Specifically, the authors analyze:

   • The Eisenhart lift, which realizes trajectories as geodesics.
   • Scalar, diagonal, and non-diagonal auxiliary lifts, where auxiliary coordinates and momenta are introduced.
   • Systems with magnetic interactions and polynomial Hamiltonians.
     In these constructions, the auxiliary momenta (or combinations thereof) form systems of particular integrals. The original dynamics are recovered by restricting to the zero level set of these momenta and projecting away the auxiliary variables.

Key Contributions and Results

• Invariant Submanifolds from Linear Closure: The paper establishes that invariant submanifolds need not be defined by global constants of motion or single scalar particular integrals. They can arise as the common zero sets of families of functions whose time derivatives close linearly on the family itself.
• Particular Liouville Integrability: The authors introduce a new notion termed "particular Liouville integrability." A system is particularly Liouville integrable if the dynamics, when restricted by a set of involutive particular integrals and projected along the characteristic distribution, yields a completely Liouville integrable Hamiltonian system. This distinguishes the result from "particular Lie integrability" (where restricted equations are solvable by quadratures but not necessarily Hamiltonian).
• Geometric Bridge: The work provides a direct geometric bridge between particular integrals, presymplectic reduction, and lower-dimensional Hamiltonian dynamics. It clarifies that when particular integrals are in involution, the reduction leads to a standard Hamiltonian system on the quotient space.
• Lift Mechanisms: The paper demonstrates that auxiliary lifts (scalar, diagonal, non-diagonal) naturally generate systems of particular integrals. In these lifted systems, the original first integrals of the base system generally cease to be global constants of motion; instead, they become part of systems of particular integrals (e.g., $f-c$ and the auxiliary momenta $p_z$) that define the invariant submanifold where the original dynamics is recovered.
• Extension to Magnetic and Polynomial Systems: The reduction mechanism is shown to extend to Hamiltonian systems with magnetic vector potentials and to polynomial Hamiltonians, provided the evolution of auxiliary momenta closes on the ideal generated by those momenta.

Significance and Claims
The paper claims to offer a "geometrically robust" form of conservation that organizes Hamiltonian dynamics even when global conservation laws are absent. Its primary significance lies in:

1. Unifying Framework: It places weak and restricted forms of integrability (where integrability holds only on selected energy or momentum levels) into a common framework with standard Liouville integrability.
2. Mechanism for Reduction: It identifies particular integrals as a generator for Hamiltonian reduction, distinct from Noether symmetries or canonical transformations.
3. Clarification of Integrability Types: It rigorously distinguishes between restricted dynamics that are merely solvable by quadratures (Lie integrability) and those that retain a Hamiltonian structure on the reduced space (Liouville integrability).

The authors note that while the construction fits naturally with the study of symmetries and conservation laws, the classification of all auxiliary lifts that transform first integrals into particular integrals remains an open direction. Similarly, extensions to Poisson, cosymplectic, and time-dependent systems are suggested as future work but are not developed in this paper.