Magnetic Geodesic Flows

• Magnetic geodesic flows are dynamical systems describing charged particle motion on Riemannian manifolds influenced by both a metric and a closed magnetic 2-form.
• The analysis on odd-dimensional spheres reveals explicit energy thresholds, connectivity conditions, and invariant submanifolds that govern the transition between supercritical and subcritical regimes.
• Superintegrability combined with U(n+1) symmetry produces explicit invariant structures and integrals, enabling detailed geometric, variational, and symplectic investigations.

Magnetic geodesic flows are dynamical systems describing the evolution of a charged particle on a Riemannian manifold under the joint influence of the metric and a prescribed magnetic field, typically represented by a closed 2-form. On odd-dimensional spheres endowed with the standard round metric and the magnetic potential given by the standard contact form, the geometry and dynamics of such flows display rich structure, including explicit thresholds for global connectivity, explicit invariant submanifolds, and a superintegrable regime. The system exhibits deep symmetries and a fine classification of invariant structures, making it an archetype for paper in geometric, variational, and symplectic aspects of magnetic geodesic phenomena (Albers et al., 4 Mar 2025).

1. Mañé Critical Value and Its Computation

A central concept in the dynamics of magnetic geodesic flows is the Mañé critical value, $c(M, g, \sigma)$, which for magnetic Lagrangian systems (geodesic flows magnetically perturbed by a closed 2-form $\sigma$) provides a sharp threshold distinguishing between "supercritical" and "subcritical" regimes. Supercritical energies ($k > c(M, g, \sigma)$) guarantee that any two points on the manifold can be joined by a magnetic geodesic of energy $k$, whereas for $k \leq c(M, g, \sigma)$ such connectivity may be lost.

On the standard sphere $S^{2n+1} \subset \mathbb{C}^{n+1}$ with round metric $g = \mathrm{Re} \langle \cdot, \cdot \rangle$ and standard contact form $$\alpha_z = \frac{1}{2} \mathrm{Re}\langle i z , \cdot \rangle$$, the Mañé critical value can be computed explicitly due to the K-contact structure (the metric dual of $\alpha$ is $(1/4)R_z$ with Reeb field $R_z = 2 i z$). The formula is

$$c(S^{2n+1}, g, \alpha) = \frac{1}{2} \|\alpha\|_\infty^2 = \frac{1}{8},$$

and the Mather set, i.e., the support of action-minimizing invariant measures at this threshold, is given by

$$\widetilde{\mathcal{M}} = \{ (z, R_z/4) : z \in S^{2n+1} \}.$$

This value demarcates the energy regime in which the global geometric and dynamical properties of the magnetic flow undergo a qualitative shift.

2. Magnetic Geodesic Connectivity and Explicit Flow Properties

The round metric and standard contact form on $S^{2n+1}$ define a canonical magnetic structure whose Lorentz force is given by $Y_z = P_z \circ i$, with $P_z$ the orthogonal projection onto $T_z S^{2n+1}$. The main result is that for energies $k > c(S^{2n+1}, g, \alpha) = 1/8$, the system satisfies a strong form of the Hopf–Rinow property: for any $q_0, q_1 \in S^{2n+1}$, there exists a magnetic geodesic of energy $k$ connecting $q_0$ to $q_1$. At or below the critical value, only specific pairs of points can be connected, determined by restrictions on the Hermitian inner product $\langle q_0, q_1 \rangle$ and the allowed "angular distance" compatible with energy constraints. The analysis thus provides a precise geometric characterization of paths and their limitations across the entire energy spectrum.

3. Magnetomorphisms and Totally Magnetic Submanifolds

The symmetry group of the system, termed the group of "magnetomorphisms," consists of diffeomorphisms that preserve both the Riemannian metric and the magnetic 2-form (or potential). In the $S^{2n+1}$ case with the contact form $\alpha$, the group is $U(n+1)$: $\text{Mag}(S^{2n+1}, g, \alpha) = U(n+1).$ A closed submanifold $N \subset S^{2n+1}$ is called totally magnetic if every magnetic geodesic initially tangent to $N$ remains (locally) inside $N$. The infinitesimal characterization of this property involves the "magnetic second fundamental form": $II^{\rm mag}_q(v) = II_q(v) + (Y^M_q v - Y^N_q v),$ where $II_q$ is the standard second fundamental form and $Y^M, Y^N$ are the respective Lorentz force fields on the ambient space and on $N$. Every connected, positive-dimensional totally magnetic submanifold is, up to $U(n+1)$-symmetry, an embedded sphere of odd dimension (intersection with a complex linear subspace), endowing the dynamics with a precise invariant stratification.

4. Superintegrability, Foliations, and Symmetries

The system is superintegrable: beyond energy conservation, there is a maximal family of independent integrals coming from the action of $U(n+1)$ and the geometry of the contact structure. Every non-Reeb magnetic geodesic is contained in a totally magnetic $3$-sphere (the intersection of $S^{2n+1}$ with a complex $2$-plane), and the flow on such a $3$-sphere is completely foliated by two-dimensional Clifford tori. The motion on these tori is quasi-periodic with a rotation number determined by the initial data ("contact angle"). The full parameterization of magnetic geodesic orbits exploits the symmetry group to reduce their paper to explicit orbits in low-dimensional invariant submanifolds.

Symmetries also control the first integrals; in particular, the central element $i \in \mathfrak{u}(n+1)$ defines at each point the "contact angle" integral,

$g_z(i z, v) = \cos \psi,$

where $\psi$ is preserved along the flow and functionally distinguishes among orbits.

5. The Contact Angle Integral: Interpolating Between Regimes

A distinguished role is played by the integral of the contact angle $\psi$, defined by $\cos \psi = g_z(i z, v)$. This classifies orbits:

• $\psi = 0, \pi$: $v$ collinear with $R_z$, i.e., the orbit is a (reparameterized) Reeb trajectory.
• $\psi = \pi/2$: $v$ tangent to the contact distribution $\ker \alpha$, so the geodesic coincides with a sub-Riemannian geodesic on $(S^{2n+1}, g, \ker \alpha)$.

More generally, the magnetic geodesic flow can be parameterized by $\psi$ and realized as a continuous interpolation between the Reeb flow and the sub-Riemannian geodesic flow: $$\gamma(t) = e^{i(s/2)t} \left( e^{-i|C_s(\psi)| t} w_0 + e^{i|C_s(\psi)| t} w_1 \right),$$ where $C_s(\psi) = (-s/2 + \cos\psi, \sin\psi)$ and $(w_0, w_1)$ is an orthonormal admissible pair. This explicitly connects the magnetic, Reeb, and sub-Riemannian regimes in a parametric family governed by the constant of motion $\psi$.

6. Implications and Broader Context

This explicit classification and control via energy and invariant quantities provide a model instance of magnetic geodesic flows with maximal symmetry and integrable structure. The explicit formula for the Mañé critical value $c(S^{2n+1}, g, \alpha) = 1/8$ offers concrete thresholds for geometric and dynamical transitions, and the identification of totally magnetic submanifolds and their invariance properties introduces a precise geometric language paralleling the notion of totally geodesic submanifolds in Riemannian geometry.

The methodology—blending explicit geometric computation, symmetry reduction, and variational classification—has implications for broader studies of magnetic flows, especially in contact and symplectic manifolds with high symmetry, and opens the possibility for analogous analysis in other settings, such as sub-Riemannian, CR, or even infinite-dimensional cases with additional structure (Albers et al., 4 Mar 2025).

7. Key Formulas and Classification Table

Concept	Explicit Formula / Group	Role in Dynamics/Symmetry
Mañé critical value	$c(S^{2n+1}, g, \alpha)=1/8$	Energy threshold for global connectivity
Reeb vector field	$R_z = 2i z$	Generates Hopf (Reeb) flow
Contact form	$$\alpha_z = \frac{1}{2} \mathrm{Re} \langle i z, \cdot \rangle$$	Defines contact distribution / Lorentz force
Magnetomorphisms	$U(n+1)$	Group of diffeomorphisms preserving (g, $\alpha$)
Contact angle	$\cos\psi = g_z(i z, v)$	Constant of motion; classifies flow regime

This canonical example of magnetic geodesic flows synthesizes explicit computation, structural invariants, and deep symmetry, providing a geometric and dynamical case paper for research in magnetic and contact flows on high-symmetry manifolds.