Scaling Symmetries and Conformal Relative Equilibria on Poisson Manifolds, with Applications to Lie-Poisson Systems
- 1Wilfrid Laurier University, Canada
Abstract
We investigate conformal relative equilibria for Hamiltonian systems on exact Poisson manifolds equipped with scaling symmetries. By introducing conformally Poisson actions and conformal momentum maps, we characterize these equilibria through an augmented Hamiltonian formulation; in the nondegenerate case, this recovers the conditions recently developed for the exact symplectic case. Specializing to the Lie-Poisson manifold g* associated with an arbitrary finite-dimensional real Lie algebra g, we show that its natural scaling action makes g* an exact Poisson manifold. We then establish a purely algebraic existence criterion: there exists a homogeneous Hamiltonian on g* admitting a nontrivial conformal relative equilibrium if and only if g contains an element whose adjoint operator has a nonzero real eigenvalue. Applying this criterion to the Bianchi classification, we identify exactly the real three-dimensional Lie algebras for which there exists a homogeneous Hamiltonian admitting a nontrivial conformal relative equilibrium. As a prominent application, we show that nontrivial conformal relative equilibria exist for a rigid body on so(2,1)*, whereas the classical free rigid body on so(3)* admits no nontrivial conformal relative equilibria.
Versions
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BibTeX
@article{santoprete2026scaling,
title = {{Scaling Symmetries and Conformal Relative Equilibria on Poisson Manifolds, with Applications to Lie-Poisson Systems}},
author = {Manuele Santoprete},
journal = {Symmetry, Integrability and Geometry: Methods and Applications},
year = {2026},
month = nov,
doi = {10.3842/sigma.2026.108},
url = {https://sigma-journal.com/2026/108/},
note = {CC BY-SA 4.0},
}
RIS
TY - JOUR
TI - Scaling Symmetries and Conformal Relative Equilibria on Poisson Manifolds, with Applications to Lie-Poisson Systems
AU - Manuele Santoprete
PY - 2026
DA - 2026/11/04
JO - Symmetry, Integrability and Geometry: Methods and Applications
DO - 10.3842/sigma.2026.108
UR - https://sigma-journal.com/2026/108/
AB - We investigate conformal relative equilibria for Hamiltonian systems on exact Poisson manifolds equipped with scaling symmetries. By introducing conformally Poisson actions and conformal momentum maps, we characterize these equilibria through an augmented Hamiltonian formulation; in the nondegenerate case, this recovers the conditions recently developed for the exact symplectic case. Specializing to the Lie-Poisson manifold g^* associated with an arbitrary finite-dimensional real Lie algebra g, we show that its natural scaling action makes g^* an exact Poisson manifold. We then establish a purely algebraic existence criterion: there exists a homogeneous Hamiltonian on g^* admitting a nontrivial conformal relative equilibrium if and only if g contains an element whose adjoint operator has a nonzero real eigenvalue. Applying this criterion to the Bianchi classification, we identify exactly the real three-dimensional Lie algebras for which there exists a homogeneous Hamiltonian admitting a nontrivial conformal relative equilibrium. As a prominent application, we show that nontrivial conformal relative equilibria exist for a rigid body on so(2,1)^*, whereas the classical free rigid body on so(3)^* admits no nontrivial conformal relative equilibria.
SN - 1815-0659
ER -