Conformal Killing vector fields and a virial theorem

2021

Phys. Rev. D

We present a new action which reproduces the cosmological sector of general relativity in both the Friedmann-Lemaître-Robertson-Walker (FLRW) and Bianchi models. This action makes no reference to the scale factor, and is of a frictional type first examined by Herglotz. We demonstrate that the extremization of this action reproduces the usual dynamics of physical observables, and the symplectification of this action is the Einstein-Hilbert action for cosmological models. We end by discussing some of the increased explanatory power produced by considering the reduced physical ontology resulting from eliminating scale.

Section: Revealing the Contact System From The Einstein-hilbert Actionmentioning

confidence: 99%

New action for cosmology

2021

Phys. Rev. D

“…This can all be generalized to include time dependence and higher derivatives, but to do so would introduce unnecessary clutter to our presentation.) In this section, we will only present a brief account of the standard results, full proofs of which can be found in [19][20][21][22]. We consider a physical system defined on the tangent bundle over a configuration space, M = TC, which is typically written in terms of positions q i and their velocities,q i .…”

Section: Lagrangian and Herglotz Mechanicsmentioning

confidence: 99%

Scale Symmetry and Friction

2021

Symmetry

Dynamical similarities are non-standard symmetries found in a wide range of physical systems that identify solutions related by a change of scale. In this paper, we will show through a series of examples how this symmetry extends to the space of couplings, as measured through observations of a system. This can be exploited to focus on observations that can be used to distinguish between different theories and identify those which give rise to identical physical evolutions. These can be reduced into a description that makes no reference to scale. The resultant systems can be derived from Herglotz’s principle and generally exhibit friction. Here, we will demonstrate this through three example systems: the Kepler problem, the N-body system and Friedmann–Lemaître–Robertson–Walker cosmology.

“…In previous work some of the basic motivation behind dynamical similarity was examined as an extension of this to a a non-strictly canonical transformation [5,6] f : Γ → Γ under which f * ω = aω for a ∈ R, a is known as the valence of the transformation. The vector field generating this on phase space is also referred to as a 'Liouville vector field' in the literature [7] 2 .…”

Section: Generating Dynamical Similaritiesmentioning

confidence: 99%

Dynamical similarity

2018

Phys. Rev. D

We examine "dynamical similarities" in the Lagrangian framework. These are symmetries of an intrinsically determined physical system under which observables remain unaffected, but the extraneous information is changed. We establish three central results in this context: i) Given a system with such a symmetry there exists a system of invariants which form a subalgebra of phase space, whose evolution is autonomous; ii) this subalgebra of autonomous observables evolves as a contact system, in which the friction-like term describes evolution along the direction of similarity; iii) the contact Hamiltonian and one-form are invariants, and reproduce the dynamics of the invariants. As the subalgebra of invariants is smaller than phase space, dynamics is determined only in terms of this smaller space. We show how to obtain the contact system from the symplectic system, and the embedding which inverts the process. These results are then illustrated in the case of homogeneous Lagrangians, including flat cosmologies minimally coupled to matter; the n-body problem and homogeneous, anisotropic cosmology.