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Urs Frauenfelder

Publications and source records attributed to Urs Frauenfelder.

At least 19 recordsLinked to original sources
arXiv2026

Merry-go-round and time-dependent symplectic forms

In the merry-go-round fictitious forces are acting like centrifugal force and Coriolis force. Like the Lorentz force Coriolis force is velocity dependent and, following Arnold, can be modeled by twisting the symplectic form. If the merry-go-round is accelerated an additional fictitious force shows up, the Euler force. In this article we explain how one deals symplectically with the Euler force by considering time-dependent symplectic forms. It will turn out that to treat the Euler force one also needs time-dependent primitives of the time-dependent symplectic forms.

math.SG↗
arXiv2026

Towards a Floer theory for Mars II -- Floer Hessian field almost extends

In part I, \cite{Frauenfelder:2026c}, we showed that collisional periodic orbits of twisted Zeeman systems can be detected variationally by a non-local Hamiltonian action functional. In this part II we show that the linearized gradient flow of this non-local functional is a Fredholm operator and prove a non-local elliptic regularity result. These results are obtained with the theory of almost extendability of weak Hessian fields introduced in \cite{Frauenfelder:2025g}.

math.SG↗
arXiv2026

Towards a Floer theory for Mars I -- Twisted Zeeman systems

In this article we study periodic orbits of an electron attracted by a proton subject to Lorentz, electric, and Euler forces where each of them is allowed to depend periodically on time. This setup is motivated by the elliptic restricted three-body-problem where the Lorentz force corresponds to Coriolis force, the Coulomb force is replaced by the gravitational force, and the electric force of an external source is a combination of centrifugal forces and gravitational forces of other bodies. This is a singular version of a Euler-Hamilton system as discussed in [FW26b]. The singularity is due to collisions of the electron with the proton, respectively of two masses. Due to the possibility of collisions this problem has to be regularized. We show how periodic collisional solutions of this problem can be detected variationally in a non-local Lagrangian setup as well as in a non-local Hamiltonian setup.

math.SG↗
arXiv2026

The Linearized Floer Equation in a Chart

In this article, we are considering the Hessian of the area functional in a non-Darboux chart. This does not seem to have been considered before and leads to an interesting new mathematical structure which we introduce in this article and refer to as almost extendable weak Hessian field. Our main result is a Fredholm theorem for Robbin-Salamon operatorsassociated to non-continuous Hessians which we prove by taking advantage of this new structure.

math.SG↗
arXiv2025

Nondegeneracy and integral count of frozen planet orbits in helium

We study a family of action functionals whose critical points interpolate between frozen planet orbits for the helium atom with mean interaction between the electrons and the free fall. The rather surprising first result of this paper asserts that for the whole family, critical points are always nondegenerate. This implies that the frozen planet orbit with mean interaction is nondegenerate and gives a new proof of its uniqueness. As an application, we show that the integral count of frozen planet orbits with instantaneous interaction equals one. For this, we prove orientability of the determinant line bundle over the space of selfadjoint Fredholm operators with spectrum bounded from below, and use it to define an integer valued Euler characteristic for Fredholm sections whose linearization belongs to this class.

math.CA↗
arXiv2025

A variational approach to frozen planet orbits in helium

We present variational characterizations of frozen planet orbits for the helium atom in the Lagrangian and the Hamiltonian picture. They are based on a Levi-Civita regularization with different time reparametrizations for the two electrons and lead to nonlocal functionals. Within this variational setup, we deform the helium problem to one where the two electrons interact only by their mean values and use this to deduce the existence of frozen planet orbits.

math.CA↗
arXiv2025

Wrapped Floer homology and the circular restricted three-body problem

Using the wrapped Floer homology, we prove the existence of consecutive collisions at the primaries in the circular restricted three-body problem. We also prove the existence of a symmetric periodic orbit. These existence results are obtained for energy hypersurfaces slightly above the first critical value.

math.SG↗
arXiv2025

A variational approach to time-dependent planar two-center Stark-Zeeman systems

We study periodic orbits in a time-dependent two-center Stark-Zeeman system, which models the motion of a charged particle attracted by two fixed Coulomb centers and subject to external magnetic and time-dependent electric fields. A motivating example is provided by the bicircular restricted four-body problem which investigates the motion of a massless particle, influenced by the Newtonian gravitational attraction of the earth, moon and periodically moving sun. Due to singularities at the Coulomb centers, standard local variational approaches fail. To overcome this, we employ the Birkhoff regularization map and construct a non-local regularized action functional on a blown-up loop space, following a recent method due to Barutello-Ortega-Vernizi \cite{BOV21}. We show that the critical points of this regularized action functional satisfy a certain second-order delay differential equation and correspond to periodic solutions of the original system, including collisions. Additionally, we examine the symmetric structure of the regularized functional.

math.SG↗
arXiv2025

Hilbert manifold structures on path spaces

In Floer theory one has to deal with two-level manifolds like for instance the space of $W^{2,2}$ loops and the space of $W^{1,2}$ loops. Gradient flow lines in Floer theory are then trajectories in a two-level manifold. Inspired by our endeavor to find a general setup to construct Floer homology we therefore address in this paper the question if the space of paths on a two-level manifold has itself the structure of a Hilbert manifold. In view of the two topologies on a two-level manifold it is unclear how to define the exponential map on a general two-level manifold. We therefore study a different approach how to define charts on path spaces of two-level manifolds. To make this approach work we need an additional structure on a two-level manifold which we refer to as tameness. We introduce the notion of tame maps and show that the composition of tame is tame again. Therefore it makes sense to introduce the notion of a tame two-level manifold. The main result of this paper shows that the path spaces on tame two-level manifolds have the structure of a Hilbert manifold.

math.SG↗
arXiv2025

The two-boost problem and Lagrangian Rabinowitz Floer homology

The two-boost problem in space mission design asks whether two points of phase space can be connected with the help of two boosts of given energy. We provide a positive answer for a class of systems related to the restricted three-body problem by defining and computing its Lagrangian Rabinowitz Floer homology. The main technical work goes into dealing with the noncompactness of the corresponding energy hypersurfaces.

math.SG↗
arXiv2025

Periodic orbits in time-dependent planar Stark-Zeeman systems

Time-dependent Stark-Zeeman systems describe the motion of an electron attracted by a proton subject to a magnetic and a time-dependent electric field. For instance the study of the dynamics of a gateway around the moon which is subject to the joint attraction of the moon, the earth and the sun leads to time-dependent Stark-Zeeman systems. In the time-dependent case there is no preserved energy. Therefore collisions cannot be regularized by blowing up the energy hypersurface. A new regularization technique of blowing up instead of the energy hypersurface the loop space was recently discovered by Barutello, Ortega, and Verzini. In this article we explain how this new regularization technique can be applied to the study of periodic orbits in time-dependent planar Stark-Zeeman systems. Since the regularization by blowing-up the loop space is nonlocal the regularized periodic orbits will not satisfy an ODE anymore but a delay equation.

math.SG↗
arXiv2025

Floerfolds and Floer functions

In this article we introduce the notion of Floer function which has the property that the Hessian is a Fredholm operator of index zero in a scale of Hilbert spaces. Since the Hessian has a complicated transformation under chart transition, in general this is not an intrinsic condition. Therefore we introduce the concept of Floerfolds for which we show that the notion of Floer function is intrinsic.

math.SG↗
arXiv2024

On the spectral flow theorem of Robbin-Salamon for finite intervals

In this article we consider operators of the form $\partial_sξ+A(s)ξ$ where $s$ lies in an interval $[-T,T]$ and $s\mapsto A(s)$ is continuous. Without boundary conditions these operators are not Fredholm. However, using interpolation theory one can define suitable boundary conditions for these operators so that they become Fredholm. We show that in this case the Fredholm index is given by the spectral flow of the operator path $A$.

math.SG↗
arXiv2024

On Kepler's geometric approach to consonances

Kepler's thinking is highly original and the inspiration for discovering his famous third law is based on his rather curious geometric approach in his Harmonices mundi for explaining consonances. In this article we try to use a modern mathematical approach based on Kepler's ideas how to characterize the seven consonances with the help of the numbers of edges of polygons constructible by ruler and compass.

math.HO↗
arXiv2024

A variational characterization of Einstein-Brillouin-Keller quantization

In this paper we explain how to construct the EBK spectrum from the marked action spectrum and derive a minimax formula for concave toric domains. In the special case of the billiard on the disk we show that while the action spectrum is algebraic the EBK spectrum has infinite transcendence degree under the assumption that Schanuel's conjecture is true.

math.SG↗