Hamiltonian perturbation theory for ultra-differentiable functions

Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard implicit function theorems. They comprise Gevrey functions and thus, as a limiting case, analytic functions. Using majorizing series, we manage to characterize them in terms of a real sequence M M bounding the growth of derivatives. In this functional setting, we prove two fundamental results of Hamiltonian perturbation theory: the invariant torus theorem, where the invariant torus remains ultra-differentiable under the assumption that its frequency satisfies some arithmetic condition which we call BR M _M , and which generalizes the Bruno-Rüssmann condition; and Nekhoroshev’s theorem, where the stability time depends on the ultra-differentiable class of the pertubation, through the same sequence M M . Our proof uses periodic averaging, while a substitute for the analyticity width allows us to bypass analytic smoothing. We also prove converse statements on the destruction of invariant tori and on the existence of diffusing orbits with ultra-differentiable perturbations, by respectively mimicking a construction of Bessi (in the analytic category) and Marco-Sauzin (in the Gevrey non-analytic category). When the perturbation space satisfies some additional condition (we then call it matching), we manage to narrow the gap between stability hypotheses (e.g. the BR M _M condition) and instability hypotheses, thus circumbscribing the stability threshold. The formulas relating the growth M M of derivatives of the perturbation on the one hand, and the arithmetics of robust frequencies or the stability time on the other hand, bring light to the competition between stability properties of nearly integrable systems and the distance to integrability. Due to our method of proof using width of regularity as a regularizing parameter, these formulas are closer to optimal as the the regularity tends to analyticity.

Separatrix divergence of stellar streams in galactic potentials

ABSTRACT Flattened axisymmetric galactic potentials are known to host minor orbit families surrounding orbits with commensurable frequencies. The behaviour of orbits that belong to these orbit families is fundamentally different than that of typical orbits with non-commensurable frequencies. We investigate the evolution of stellar streams on orbits near the boundaries between orbit families (separatrices) in a flattened axisymmetric potential. We demonstrate that the separatrix divides these streams into two groups of stars that belong to two different orbit families, and that as a result, these streams diffuse more rapidly than streams that evolve elsewhere in the potential. We utilize Hamiltonian perturbation theory to estimate both the time-scale of this effect and the likelihood of a stream evolving close enough to a separatrix to be affected by it. We analyse two prior reports of stream-fanning in simulations with triaxial potentials, and conclude that at least one of them is caused by separatrix divergence. These results lay the foundation for a method of mapping the orbit families of galactic potentials using the morphology of stellar streams. Comparing these predictions with the currently known distribution of streams in the Milky Way presents a new way of constraining the shape of our Galaxy’s potential and distribution of dark matter.

Spacelike deformations: higher-helicity fields from scalar fields

In contrast to Hamiltonian perturbation theory which changes the time evolution, “spacelike deformations” proceed by changing the translations (momentum operators). The free Maxwell theory is only the first member of an infinite family of spacelike deformations of the complex massless Klein–Gordon quantum field into fields of higher helicity. A similar but simpler instance of spacelike deformation allows to increase the mass of scalar fields.