Abstract
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.
Highlights
We have demonstrated several results at the heart of shape dynamics
These are: (i) scaling symmetries between theories can be expressed as dynamical similarities by extending our description to include the coupling constants of the theories as velocities
This allowed us to show that there is a description of systems which have such symmetries which makes no reference to scale. (ii) The evolution of these systems can be expressed entirely in terms of ‘shapes’
Summary
One might object to the rescaling in the time-one system, as it would evolve more slowly in t than the other This time must be read from a physical device;qif we equip our observer with a pendulum clock, for example, with time period T = 2π l , to follow Poincaré, we must set l = λl g for all lengths to be affected and g derives from the radius of the Earth, its mass and the Newton potential. Rescaling all of these (keeping the density of the Earth fixed), we see that the clock would have the time period T = λT.
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