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Abstract

This thesis investigates the geometry and dynamics of magnetic systems on both finite- and infinite-dimensional manifolds, and their interplay with symplectic capacities and geometric hydrodynamics.

In finite dimensions, we study magnetic geodesics on odd-dimensional spheres equipped with the round metric and standard contact form. We compute the Mañé critical value and show that an energy level is supercritical if and only if all pairs of points on the sphere can be connected by a magnetic geodesic of that energy. Along the way, we introduce the notions of totally magnetic submanifolds and magnetomorphisms.

Extending these techniques, we study magnetic billiards induced by this setting and compute the Hofer-Zehnder capacity of disk tangent bundles of lens spaces. This reveals an unexpected insensitivity to the covering space structure and provides the first example of disk tangent bundles where the Gromov width and the Hofer-Zehnder capacity differ. Our methods combine magnetic billiards for the lower bound with Gromov-Witten invariants for the upper bound.

We also resolve—for a broad class of magnetic systems—a question posed by Contreras, Iturriaga, Paternain, and Paternain in 1998, known as the contact type conjecture. To this end, we explicitly construct, on any closed manifold, an infinite-dimensional space of exact magnetic systems, which we call magnetic systems of strong geodesic type. For these systems, each energy level below the Mañé threshold contains null-homologous periodic orbits of negative action, thereby proving non-contact type behavior and confirming the conjecture in this context. We obtain explicit formulas for Mañé’s critical values, as well as multiplicity results that establish the abundance of periodic magnetic geodesics.

In the infinite-dimensional setting, we consider a Lie group equipped with the H^1-dot metric and a contact-type magnetic field, such that the corresponding magnetic geodesic equation is the magnetic two-component Hunter-Saxton system (M2HS), a nonlinear PDE system. We define and compute the Mañé critical value in this framework and show that an energy level is supercritical if and only if every pair of points on the sphere can be connected by a magnetic geodesic at that energy.

Through this geometric perspective, we further obtain analytical results, including a blow-up analysis for (M2HS) and the existence of global conservative weak solutions. A key ingredient is the observation that the Madelung transform can be interpreted as a magnetomorphism, embedding the system into a flow on an infinite-dimensional sphere, which reduces further to a totally magnetic three-sphere.

Finally, we combine V. Arnold’s celebrated approach via the Euler-Arnold equation—which interprets the incompressible Euler equations as geodesic flow on a Lie group with a right-invariant metric—with his formulation of the motion of a charged particle in a magnetic field. This leads us to introduce the magnetic Euler-Arnold equation, the Eulerian form of the magnetic geodesic flow on a Lie group endowed with a right-invariant metric and a right-invariant closed two-form serving as the magnetic field.

As an illustration, we show that the Korteweg-de Vries equation, the generalized Camassa-Holm equation, the infinite conductivity equation, and the global quasi-geostrophic equations can all be interpreted as magnetic Euler-Arnold equations.