Dynamical Similarity

Abstract

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.

17 pages. Updated to remove a flaw in section II and align with standard terminology

References in corpus (4)

• Einstein gravity as a 3D conformally invariant theory
• On the measure problem in slow roll inflation and loop quantum cosmology
• Eternal Universe
• Liouville's Theorem and the canonical measure for nonconservative systems from contact geometry

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• Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions
• Deterministic evolution of gauge fields through a singularity
• New directions for contact integrators
• Relative Equilibria for Scaling Symmetries and Central Configurations