When scale is surplus

We study a long-recognised but under-appreciated symmetry called dynamical similarity and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a system where the unit of Hamilton’s principal function is rescaled, and therefore represent a kind of dynamical scaling symmetry with formal properties that differ from many standard symmetries. To study this symmetry, we develop a general framework for symmetries that distinguishes the observable and surplus structures of a theory by using the minimal freely specifiable initial data for the theory that is necessary to achieve empirical adequacy. This framework is then applied to well-studied examples including Galilean invariance and the symmetries of the Kepler problem. We find that our framework gives a precise dynamical criterion for identifying the observables of those systems, and that those observables agree with epistemic expectations. We then apply our framework to dynamical similarity. First we give a general definition of dynamical similarity. Then we show, with the help of some previous results, how the dynamics of our observables leads to singularity resolution and the emergence of an arrow of time in cosmology.

The dynamically selected stellar halo of the Galaxy with Gaia and the tilt of the velocity ellipsoid

Aims. We study the dynamical properties of halo stars located in the solar neighbourhood. Our goal is to explore how the properties of the halo depend on the selection criteria used to define a sample of halo stars. Once this is understood, we proceed to measure the shape and orientation of the halo’s velocity ellipsoid and we use this information to put constraints on the gravitational potential of the Galaxy. Methods. We use the recently released Gaia DR1 catalogue cross-matched to the RAVE dataset for our analysis. We develop a dynamical criterion based on the distribution function of stars in various Galactic components, using action integrals to identify halo members, and we compare this to the metallicity and to kinematically selected samples. Results. With this new method, we find 1156 stars in the solar neighbourhood that are likely members of the stellar halo. Our dynamically selected sample consists mainly of distant giants on elongated orbits. Their metallicity distribution is rather broad, with roughly half of the stars having [M/H] ≥ −1 dex. The use of different selection criteria has an important impact on the characteristics of the velocity distributions obtained. Nonetheless, for our dynamically selected and for the metallicity selected samples, we find the local velocity ellipsoid to be aligned in spherical coordinates in a Galactocentric reference frame. This suggests that the total gravitational potential is rather spherical in the region spanned by the orbits of the halo stars in these samples.

The locally translation structure of flows on the wandering points set

A dynamical criterion of conjugacy of a flow with a shift and a counterpart of the Brouwer plane translation theorem for flows on a normal topological space are established.