Reduction for constrained variational problems on 3D null curves

We consider the optimal control problem for null curves in de Sitter 3-space defined by a functional which is linear in the curvature of the trajectory. We show how techniques based on the method of moving frames and exterior differential systems, coupled with the reduction procedure for systems with a Lie group of symmetries lead to the integration by quadratures of the extremals. Explicit solutions are found in terms of elliptic functions and integrals.Comment: 16 page

Björling type problems for elastic surfaces

In this survey we address the Bj"orling problem for various classes of surfaces associated to the Euler--Lagrange equation of the Helfrich elastic energy subject to volume and area constraints

The interaction between stray electrostatic fields and a charged free-falling test mass

We present an experimental analysis of force noise caused by stray electrostatic fields acting on a charged test mass inside a conducting enclosure, a key problem for precise gravitational experiments. Measurement of the average field that couples to test mass charge, and its fluctuations, is performed with two independent torsion pendulum techniques, including direct measurement of the forces caused by a change in electrostatic charge. We analyze the problem with an improved electrostatic model that, coupled with the experimental data, also indicates how to correctly measure and null the stray field that interacts with test mass charge. Our measurements allow a conservative upper limit on acceleration noise, of 2 fm/s$^2$\rthz\ for frequencies above 0.1 mHz, for the interaction between stray fields and charge in the LISA gravitational wave mission.Comment: Minor edits in PRL publication proces

Closed trajectories of a particle model on null curves in anti-de Sitter 3-space

We study the existence of closed trajectories of a particle model on null curves in anti-de Sitter 3-space defined by a functional which is linear in the curvature of the particle path. Explicit expressions for the trajectories are found and the existence of infinitely many closed trajectories is proved.Comment: 12 pages, 1 figur

Hyperpolarizability and operational magic wavelength in an optical lattice clock

Optical clocks benefit from tight atomic confinement enabling extended interrogation times as well as Doppler- and recoil-free operation. However, these benefits come at the cost of frequency shifts that, if not properly controlled, may degrade clock accuracy. Numerous theoretical studies have predicted optical lattice clock frequency shifts that scale nonlinearly with trap depth. To experimentally observe and constrain these shifts in an $^{171}$Yb optical lattice clock, we construct a lattice enhancement cavity that exaggerates the light shifts. We observe an atomic temperature that is proportional to the optical trap depth, fundamentally altering the scaling of trap-induced light shifts and simplifying their parametrization. We identify an "operational" magic wavelength where frequency shifts are insensitive to changes in trap depth. These measurements and scaling analysis constitute an essential systematic characterization for clock operation at the $10^{-18}$ level and beyond.Comment: 5 + 2 pages, 3 figures, added supplementa

Hamiltonian flows on null curves

The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the trajectories of a certain particle model on null curves described by a Lagrangian linear in the curvature. In addition, it is shown that the curvature of a null curve which evolves by similarities can be computed in terms of the solutions of the second Painlev\'e equation.Comment: 14 pages, v2: final version; minor changes in the expositio