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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Dynamical similarity
AU - Sloan, David James Austin
N1 - © 2018 American Physical Society
PY - 2018/6/28
Y1 - 2018/6/28
N2 - 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.
AB - 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.
U2 - 10.1103/PhysRevD.97.123541
DO - 10.1103/PhysRevD.97.123541
M3 - Journal article
VL - 97
JO - Physical Review D
JF - Physical Review D
SN - 1550-7998
M1 - 123541
ER -