2018/03/31 by David Sloan · 4 citations
Mathematics · Physics and Astronomy · #Algebra over a field #Artificial intelligence #Black Holes and Theoretical Physics #Chaotic #Classical mechanics #Computer science #Context (archaeology) #Cosmology and Gravitation Theories #Dynamical systems theory #Embedding #Geometry #Hamiltonian (control theory) #Hamiltonian system #Homogeneous space #Mathematics #Noncommutative and Quantum Gravity Theories #Observable #Phase space #Physics #Pure mathematics #Quantum mechanics #Subalgebra #Symplectic geometry #Theoretical physics #gr-qc #hep-th
paper · pdf · doi:10.1103/physrevd.97.123541
published as Phys. Rev. D 97, 123541 (2018) · 17 pages. Updated to remove a flaw in section II and align with standard terminology
arxiv created 2018/04/03 · openalex publication_date 2018/06/28 · arxiv updated 2018/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 frictionlike 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.