Deformation of Lie–Poisson algebras and chirality
Abstract
Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian symmetric spectra: If λ is an eigenvalue of the linearized generator, -λ and $$\barλ$$ (hence, -$$\barλ$$) are also eigenvalues—the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution. However, linearization around a singular equilibrium point (which commonly exists in noncanonical Hamiltonian systems) works out differently, resulting in breaking of the Hamiltonian symmetry of spectra; time-reversal asymmetry causes chirality. This interesting phenomenon was first found in analyzing the chiral motion of the rattleback, a boat-shaped top having misaligned axes of inertia and geometry [Z. Yoshida et al., Phys. Lett. A 381, 2772–2777 (2017)]. To elucidate how chiral spectra are generated, we study the three-dimensional Lie–Poisson systems and classify the prototypes of singularities that cause symmetry breaking. The central idea is the deformation of the underlying Lie algebra; invoking Bianchi’s list of all three-dimensional Lie algebras, we show that the so-called class-B algebras, which are produced by asymmetric deformations of the simple algebra , yield chiral spectra when linearized around their singularities. The theory of deformation is generalized to higher dimensions, including the infinite-dimensional Poisson manifolds relevant to fluid mechanics.
- Publication Date:
- Research Org.:
- Univ. of Texas, Austin, TX (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1800222
- Alternate Identifier(s):
- OSTI ID: 1646529
- Grant/Contract Number:
- FG02-04ER54742; FG02-04ER- 54742
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Mathematical Physics
- Additional Journal Information:
- Journal Volume: 61; Journal Issue: 8; Journal ID: ISSN 0022-2488
- Publisher:
- American Institute of Physics (AIP)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics; Hamiltonian mechanics; Differential geometry; Vortex dynamics; Differential topology; Lie algebras; Fluid mechanics; Associative algebra; Chirality; Differentiable manifold
Citation Formats
Yoshida, Zensho, and Morrison, Philip J. Deformation of Lie–Poisson algebras and chirality. United States: N. p., 2020.
Web. doi:10.1063/1.5145218.
Yoshida, Zensho, & Morrison, Philip J. Deformation of Lie–Poisson algebras and chirality. United States. https://doi.org/10.1063/1.5145218
Yoshida, Zensho, and Morrison, Philip J. Thu .
"Deformation of Lie–Poisson algebras and chirality". United States. https://doi.org/10.1063/1.5145218. https://www.osti.gov/servlets/purl/1800222.
@article{osti_1800222,
title = {Deformation of Lie–Poisson algebras and chirality},
author = {Yoshida, Zensho and Morrison, Philip J.},
abstractNote = {Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian symmetric spectra: If λ is an eigenvalue of the linearized generator, -λ and $\barλ$ (hence, -$\barλ$) are also eigenvalues—the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution. However, linearization around a singular equilibrium point (which commonly exists in noncanonical Hamiltonian systems) works out differently, resulting in breaking of the Hamiltonian symmetry of spectra; time-reversal asymmetry causes chirality. This interesting phenomenon was first found in analyzing the chiral motion of the rattleback, a boat-shaped top having misaligned axes of inertia and geometry [Z. Yoshida et al., Phys. Lett. A 381, 2772–2777 (2017)]. To elucidate how chiral spectra are generated, we study the three-dimensional Lie–Poisson systems and classify the prototypes of singularities that cause symmetry breaking. The central idea is the deformation of the underlying Lie algebra; invoking Bianchi’s list of all three-dimensional Lie algebras, we show that the so-called class-B algebras, which are produced by asymmetric deformations of the simple algebra so( 3) , yield chiral spectra when linearized around their singularities. The theory of deformation is generalized to higher dimensions, including the infinite-dimensional Poisson manifolds relevant to fluid mechanics.},
doi = {10.1063/1.5145218},
journal = {Journal of Mathematical Physics},
number = 8,
volume = 61,
place = {United States},
year = {Thu Aug 06 00:00:00 EDT 2020},
month = {Thu Aug 06 00:00:00 EDT 2020}
}
Search WorldCat to find libraries that may hold this journalWeb of Science
Works referenced in this record:
Reduction of symplectic manifolds with symmetry
journal, February 1974
- Marsden, Jerrold; Weinstein, Alan
- Reports on Mathematical Physics, Vol. 5, Issue 1
Rattleback: A model of how geometric singularity induces dynamic chirality
journal, September 2017
- Yoshida, Z.; Tokieda, T.; Morrison, P. J.
- Physics Letters A, Vol. 381, Issue 34
The modular automorphism group of a Poisson manifold
journal, November 1997
- Weinstein, Alan
- Journal of Geometry and Physics, Vol. 23, Issue 3-4
An invitation to singular symplectic geometry
journal, January 2019
- Braddell, Roisin; Delshams, Amadeu; Miranda, Eva
- International Journal of Geometric Methods in Modern Physics, Vol. 16, Issue supp01
Invariants of real low dimension Lie algebras
journal, June 1976
- Patera, J.; Sharp, R. T.; Winternitz, P.
- Journal of Mathematical Physics, Vol. 17, Issue 6
Deformations of Lie Algebra Structures
journal, January 1967
- Nijenhuis, Albert; Richardson, Jr., R.
- Indiana University Mathematics Journal, Vol. 17, Issue 1
Remarks on spectra of operator rot
journal, December 1990
- Yoshida, Zensho; Giga, Yoshikazu
- Mathematische Zeitschrift, Vol. 204, Issue 1
Deformations and contractions of Lie algebras
journal, June 2005
- Fialowski, Alice; Montigny, Marc de
- Journal of Physics A: Mathematical and General, Vol. 38, Issue 28
Hamiltonian moment reduction for describing vortices in shear
journal, August 1997
- Meacham, S. P.; Morrison, P. J.; Flierl, G. R.
- Physics of Fluids, Vol. 9, Issue 8
On the use of projectors for Hamiltonian systems and their relationship with Dirac brackets
journal, March 2013
- Chandre, C.; de Guillebon, L.; Back, A.
- Journal of Physics A: Mathematical and Theoretical, Vol. 46, Issue 12
Hierarchical structure of noncanonical Hamiltonian systems
journal, January 2016
- Yoshida, Z.; Morrison, P. J.
- Physica Scripta, Vol. 91, Issue 2