Right-Invariant Magnetic Systems

• Right-Invariant Magnetic Systems are geometric dynamical systems on Lie groups characterized by right-invariant Riemannian metrics and closed 2-forms acting as magnetic fields.
• They model magnetic geodesics by incorporating Lorentz forces that deform classical geodesic flows, with applications in mechanics, hydrodynamics, and integrable systems.
• These systems feature energy thresholds via Mañé's critical value and enable the analysis of variational principles and symmetry reductions in both finite and infinite-dimensional contexts.

A right-invariant magnetic system is a geometric dynamical system defined on a (finite- or infinite-dimensional) Lie group or half-Lie group, endowed with both a right-invariant Riemannian metric and a right-invariant closed 2-form that plays the role of a magnetic field. The right-invariant structure ensures that the dynamics are equivariant under group multiplication, and the magnetic field introduces a nontrivial Lorentz-type force or geometric twist to the trajectories of geodesic flow. Right-invariant magnetic systems are central in classical and quantum mechanics, hydrodynamics, geometric control theory, and the analysis of integrable systems, with significant connections to infinite-dimensional geometry, representation theory, and variational calculus.

1. Structural Foundations: Metrics, Magnetic Forms, and Dynamics

A right-invariant magnetic system on a Lie group $G$ (or a half-Lie group, e.g., groups of diffeomorphisms) consists of:

• A strong right-invariant Riemannian metric $g$: for every $x \in G$, $g_x((R_x)_* v, (R_x)_* w) = g_e(v, w)$ where $R_x$ denotes right translation by $x$ and $e$ is the identity.
• A right-invariant closed 2-form $\sigma$: $$\sigma_{x}( (R_x)_* v, (R_x)_* w ) = \sigma_e(v, w)$$.

The equations of motion are governed by the magnetic geodesic equation

$$\nabla_{\dot\gamma}\dot\gamma = Y(\gamma)\dot\gamma$$

where $\nabla$ is the Levi-Civita connection, $\gamma$ is a curve in $G$, and the Lorentz force $Y$ is defined via $g_x(Y_x u, v) = \sigma_x(u, v)$. For vanishing $\sigma$, the system reduces to the geodesic flow of $(G, g)$. The right-invariant structure ensures that all geometric and dynamical properties are preserved under right multiplication, leading to a natural symmetry group acting on the dynamics.

For infinite-dimensional half-Lie groups (where only right translations are smooth), such as certain diffeomorphism groups, the same framework applies albeit with additional analytical subtleties (Maier et al., 22 Oct 2025).

2. Mañé's Critical Value and Energy Thresholds

A central concept in the global analysis of right-invariant magnetic systems is Mañé's critical value. Suppose $\sigma$ is weakly exact (i.e., its pullback to the universal cover $\widehat{G}$ is exact: $d\hat{\alpha} = \hat{\sigma}$ for a right-invariant 1-form $\hat{\alpha}$). The Mañé critical value is

$$c(G, g, \sigma) := \inf \left\{ \frac{1}{2}\|\hat{\alpha}\|_\infty^2 : d\hat{\alpha} = \hat{\sigma},\ \hat{\alpha}\ \text{right-invariant} \right\}$$

and serves as a threshold energy.

For energies $\kappa > c(G, g, \sigma)$, magnetic geodesics with prescribed energy exhibit strong variational and dynamical properties: the lift of the magnetic flow to $\widehat{G}$ coincides with the geodesic flow of a Randers-type Finsler metric $$F^\kappa(\hat{x}, v) = \sqrt{2\kappa}|v|_{\hat{x}} - \hat{\alpha}_{\hat{x}}(v)$$ (Maier et al., 22 Oct 2025). Below or at the critical value, global existence and connectivity properties may fail.

Explicit computations arise in finite and infinite dimensions. On odd-dimensional spheres $S^{2n+1}$ with the standard contact form $\alpha$, $c(L) = \frac{1}{8}$, as the metric dual of $\alpha$ has constant length (Albers et al., 4 Mar 2025, Maier, 17 Mar 2025).

3. Hopf–Rinow Theorem and Global Geodesic Properties

The classical Hopf–Rinow theorem asserts that a complete Riemannian manifold has the property that any two points can be joined by a minimizing geodesic. In right-invariant magnetic systems, the presence of the magnetic field deforms the geodesic flow, but an analog holds: for energy levels above Mañé's critical value, for any two points in $G$ (or its universal cover), there exists a (possibly magnetic) geodesic connecting them that minimizes an appropriate action functional. This completeness and connectivity are established for infinite-dimensional half-Lie groups, generalizing results from closed finite-dimensional manifolds (Maier et al., 22 Oct 2025, Albers et al., 4 Mar 2025, Maier, 17 Mar 2025). Below the critical value, this property can fail, as in the classical case.

4. Explicit and Canonical Constructions: Integrability, Reduction, and Classification

Right-invariant magnetic systems admit both general abstract theory and detailed canonical constructions. In the finite-dimensional Lie group case, the right-invariant closed 2-form corresponds to a Lie algebra 2-cocycle and can induce nontrivial central extensions of the symmetry algebra (Magazev et al., 2011). In four dimensions, canonical forms for 2-cocycles and explicit integrability conditions are available: the system is integrable in quadratures if and only if

$$\frac{1}{2} ( \dim \mathfrak{g} - \mathrm{ind}_{[F]} \mathfrak{g} ) < 2,$$

where $F$ is the magnetic 2-form and $\mathrm{ind}_{[F]} \mathfrak{g}$ is the cohomology index of $F$.

On homogeneous spaces, gyroscopic (Chaplygin) reduction, invariant measure, and Hamiltonization can be analyzed for right-invariant magnetic systems, including classification of so-called totally magnetic submanifolds and their internal symmetries, known as magnetomorphisms (diffeomorphisms preserving both metric and magnetic form) (Dragovic et al., 2021, Albers et al., 4 Mar 2025). On Walker–Stiefel varieties $V_{n,2}$, explicit integrability for SO($n$)-invariant metrics and contact-type magnetic fields is achieved (Jovanovic, 16 Jun 2025).

5. Infinite-Dimensional Generalizations and Geometric Hydrodynamics

Extending right-invariant magnetic systems to infinite-dimensional contexts leads to significant frameworks in mathematical physics and hydrodynamics. The Euler–Arnold equation on a (possibly infinite-dimensional) Lie group $G$ with a right-invariant metric $g$, now augmented by a right-invariant closed 2-form $\sigma$, becomes the magnetic Euler–Arnold equation: $$u_t = -\operatorname{ad}^T_u(u) - Y_\mathrm{id}(u),$$ where $Y$ is defined via $$g_\mathrm{id}(Y_\mathrm{id}(u), v) = \sigma_\mathrm{id}(u, v)$$. In various limits or for certain choices of $G$, this encompasses:

• The Korteweg–de Vries equation as a magnetic deformation of Euler–Arnold flow on $\mathrm{Diff}(S^1)$ with the Gelfand–Fuchs cocycle (Maier, 31 May 2025);
• The Camassa–Holm equation as a magnetic Euler–Arnold equation for the right-invariant $H^1$ metric on $\mathrm{Diff}(S^1)$;
• The infinite conductivity and global quasi-geostrophic equations as magnetic geodesic flows on volume-preserving diffeomorphism groups or quantomorphism groups.

This geometric viewpoint provides powerful interpretive and analytic tools, permitting both the deduction of dispersion/convection terms from group-theoretic data and rigorous well-posedness results for the associated PDEs (Maier, 31 May 2025).

6. Magnetomorphisms, Totally Magnetic Submanifolds, and Dynamical Reduction

Magnetomorphisms are diffeomorphisms preserving both the metric and the magnetic 2-form. In right-invariant settings, these are often large groups—e.g., $U(n+1)$ for the round metric and standard contact form on $S^{2n+1}$ (Albers et al., 4 Mar 2025). The concept of totally magnetic submanifolds extends the notion of totally geodesic submanifolds: a submanifold $N$ is totally magnetic if any magnetic geodesic tangent to $N$ remains entirely within $N$, equivalently if a magnetic second fundamental form vanishes (Albers et al., 4 Mar 2025).

Infinite-dimensional right-invariant magnetic systems, upon reduction via a suitable generalized Madelung transform, may have all orbits confined to invariant finite-dimensional totally magnetic spheres; this "dynamical reduction theorem" drastically simplifies qualitative dynamics and permits invariants and blow-up criteria to be deduced from finite-dimensional models (Albers et al., 4 Mar 2025, Maier, 17 Mar 2025).

7. Applications, Implications, and Open Directions

Right-invariant magnetic systems are the foundation for the analysis of integrable magnetic geodesic flows, superintegrability in the presence of external fields, and the construction of quantum systems with manifest symmetries (e.g., via central extension of symmetry groups and harmonic analysis) (Davighi et al., 2019). Advanced applications include the modeling of helical undulators in solenoidal fields (Kubů et al., 2022), the classification of rigid body gyrostats (Jovanovic, 16 Jun 2025), and rigorous geometric hydrodynamics (Maier, 31 May 2025).

Crucially, their paper unifies variational principles, geometric mechanics, Lie algebra cohomology, and infinite-dimensional analysis. Open problems remain regarding the extension of Mañé’s critical value and variational theorems beyond weakly exact forms and into broader infinite-dimensional or singular settings, as well as the rigorous mathematical understanding of complex, non-Hermitian extensions in quantum superintegrable systems (Kubů et al., 2022).

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Table: Key Components of Right-Invariant Magnetic Systems

Structure	Role	Example
Right-inv. metric g	Defines kinetic energy, invariance	$g_x((R_x)_* v, (R_x)_* w) = g_e(v, w)$
Right-inv. 2-form σ	Magnetic field, Lorentz force term	σ = dα, α standard contact form
Mañé critical value	Energy threshold for connectivity	$c = \frac{1}{8}$ on $S^{2n+1}$

The theory and classification of right-invariant magnetic systems provide a robust foundation for the analysis of geometric flows, global minimization, and integrability in the presence of external (magnetic/gyroscopic) fields in both finite and infinite-dimensional geometric mechanics (Magazev et al., 2011, Albers et al., 4 Mar 2025, Maier, 17 Mar 2025, Maier, 31 May 2025, Maier et al., 22 Oct 2025).