Scaling Symmetries, Contact Reduction and Poincaré's dream

Abstract

A symplectic Hamiltonian system admitting a scaling symmetry can be reduced to an equivalent contact Hamiltonian system in which some physically-irrelevant degree of freedom has been removed. As a consequence, one obtains an equivalent description for the same physical phenomenon, but with fewer inputs needed, thus realizing 'Poincaré's dream' of a scale-invariant description of the universe.This work is devoted to a thorough analysis of the mathematical framework behind such reductions. We show that generically such reduction is possible and the reduced (fundamental) system is a contact Hamiltonian system. The price to pay for this level of generality is that one is compelled to consider the coupling constants appearing in the original Hamiltonian as part of the dynamical variables of a lifted system. This however has the added advantage of removing the hypothesis of the existence of a scaling symmetry for the original system at all, without breaking the sought-for reduction in the number of inputs needed. Therefore a large class of Hamiltonian (resp. Lagrangian) theories can be reduced to scale-invariant contact Hamiltonian (resp. Herglotz variational) theories.

Herglotz action for homogeneous cosmologies

“…x where ι indicates the interior product. Details of this construction can be found in [29]. Note the from our definitions…”

“…The frictional nature of these descriptions also provide an explanation for the evolution of physical (Liouville) measures on phase space [24][25][26][27], and see [28] for a mathematical description. Very recently, a complete mathematical description of the contact reduction has been shown for both the Lagrangian and Hamiltonian formalisms [29].…”

The flow method for the Baker-Campbell-Hausdorff formula: exact results

“…We summarize here the essential concepts of contact geometry that are needed for this work and refer the reader to [4,7,10,11,13,23,24,29,33] for additional details, including the relevant proofs. The main defining object for us is a special differential 1-form, called the contact form, which induces both a volume form on the manifold and a maximally non-integrable distribution on its tangent bundle, called the contact structure.…”

Herglotz Action for Homogeneous Cosmology

“…The frictional nature of these descriptions also provide an explanation for the evolution of physical (Liouville) measures on phase space [21][22][23][24], and see [25] for a mathematical description. Very recently, a complete mathematical description of the contact reduction has been shown for both the Lagrangian and Hamiltonian formalisms [26].…”