Multisymplectic structures and invariant tensors for Lie systems

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find methods to derive superposition rules, constants of motion, and invariant tensor fields relative to the evolution of the multisymplectic Lie system. Our results are illustrated with examples occurring in physics, mathematics, and control theory.

💡 Research Summary

The paper introduces the concept of multisymplectic Lie systems, a class of Lie systems whose associated Vessiot‑Guldberg Lie algebra consists of Hamiltonian vector fields with respect to a multisymplectic (i.e., higher‑order symplectic) form. After recalling the Lie‑Scheffers theorem and the definition of a Lie system, the authors define a compatible multisymplectic form Θ on a manifold N such that every vector field Xα in the Vessiot‑Guldberg algebra V satisfies i_{Xα}Θ = dHα for some (k‑1)‑form Hα. The triple (N, Θ, X) is then called a multisymplectic Lie system.

The core technical development relies on attaching a tensor coalgebra to the Lie algebra g that is isomorphic to V. By extending the adjoint representation of g to the full tensor algebra T(g) and its symmetric and exterior subalgebras S(g) and Λ(g), the authors turn these spaces into g‑modules. They then endow T(g) (and its sub‑modules) with a coalgebra structure, which can be iterated to form tensor products T^{(m)}(g) = T(g) ⊠ … ⊠ T(g). Within these coalgebras, g‑invariant elements such as Casimir operators (symmetric tensors) and Chevalley‑Eilenberg cocycles (antisymmetric tensors) are identified. By pulling back these invariant tensors through the representation map, one obtains invariant tensor fields on N and on the diagonal prolongations N^m of the original system. These invariant fields give rise to constants of motion and, crucially, to superposition rules for the original Lie system without solving any auxiliary partial differential equations.

A significant portion of the work is devoted to “locally automorphic” Lie systems, i.e., systems locally diffeomorphic to automorphic Lie systems (systems whose Vessiot‑Guldberg algebra acts transitively). The authors prove that when the Vessiot‑Guldberg algebra is unimodular, a compatible multisymplectic form can be constructed algebraically (Theorems 4.9, 4.11, Corollary 4.12). This provides a systematic method for endowing a broad class of Lie systems with multisymplectic structures, and consequently with the rich invariant machinery described above.

The theoretical framework is illustrated through several examples. For the Schwarzian derivative equation, the multisymplectic approach reproduces known invariants and yields a new explicit superposition formula. In Riccati‑type diffusion equations, the method supplies conserved quantities and a compact expression for the general solution. The Darboux‑Brioschi‑Halphen system, a classical integrable model, is treated to exhibit how Casimir elements of the associated Lie algebra generate the well‑known first integrals. Control‑theoretic models are also examined, showing that the multisymplectic formalism can be applied to design and analyze controllable Lie systems.

Finally, the paper discusses how multisymplectic Lie systems encompass and extend previously studied Hamiltonian, Dirac, k‑symplectic, and Jacobi Lie systems. By providing a unified algebraic‑geometric toolkit—tensor coalgebras, invariant tensors, and compatible multisymplectic forms—the authors open new avenues for the analysis of nonlinear differential equations, integrable systems, and geometric control theory, potentially impacting quantization procedures and higher‑order symmetry investigations.