Эта публикация цитируется в
1 статье
Alexey Borisov Memorial Volume
Möbius Fluid Dynamics on the Unitary Groups
Daniela Emmanuelea,
Marcos Salvaib,
Francisco Vittonea a Universidad Nacional de Rosario,
Av. Pellegrini 250, 2000 Rosario, Argentina
b FaMAF, Universidad Nacional de Córdoba; CIEM, CONICET,
Ciudad Universitaria, 5000 Córdoba, Argentina
Аннотация:
We study the nonrigid dynamics induced by the standard birational actions of
the split unitary groups
$G=O_{o}\left( n,n\right) $,
$SU\left( n,n\right) $
and
$Sp\left( n,n\right) $ on the compact classical Lie groups
$M=SO_{n}$,
$U_{n}$ and
$Sp_{n}$, respectively. More precisely, we study the geometry of
$G$ endowed with the kinetic energy metric associated with the action of
$G$
on
$M,$ assuming that
$M$ carries its canonical bi-invariant Riemannian
metric and has initially a homogeneous distribution of mass. By the least
action principle, force-free motions (thought of as curves in
$G$)
correspond to geodesics of
$G$. The geodesic equation may be understood as
an inviscid Burgers equation with Möbius constraints. We prove that the
kinetic energy metric on
$G$ is not complete and in particular not
invariant, find symmetries and totally geodesic submanifolds of
$G$ and
address the question under which conditions geodesics of rigid motions are
geodesics of
$G$. Besides, we study equivalences with the dynamics of
conformal and projective motions of the sphere in low dimensions.
Ключевые слова:
force-free motion, kinetic energy metric, nonrigid dynamics, unitary group, split
unitary group, Möbius action, maximal isotropic subspace, inviscid Burgers equation.
MSC: 22F50,
53C22,
58D19,
70K25,
70G65,
76M60 Поступила в редакцию: 23.04.2021
Принята в печать: 17.03.2022
Язык публикации: английский
DOI:
10.1134/S1560354722030054