Abstract
Using the U(2, 2)-invariant symplectic structure of the Penrose twistor space we find full and complete E(3)-equivariant symplectic realizations of some Poisson submanifolds of the Lie-Poisson space \(\mathbf {e}(3)^*\cong \mathbb {R}^3\times \mathbb {R}^3\) dual to the Lie algebra e(3) of the Euclidean group E(3), which is an underlying space in the rigid body theory. Considering concrete integrable cases of gyrostat systems on e(3)∗ we can take their liftings to the ones on the constructed symplectic realizations. This way we obtain integrable systems on the phase spaces given by the symplectic realizations. As examples we compute integrable Hamiltonian systems obtained through symplectic realizations of the Lagrange, Kovalevskaya-Yahia, and Clebsh integrable cases.
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References
Bolsinov, A.V., Fomenko, A.T.: Integrable Hamiltonian systems. Chapman & Hall/CRC, Boca Raton, FL (2004). https://doi.org/10.1201/9780203643426. Geometry, topology, classification, Translated from the 1999 Russian original
Odzijewicz, A., Sliżewska, A., Wawreniuk, E.: A family of integrable perturbed Kepler systems. Russ. J. Math. Phys. 26(3), 368–383 (2019). https://doi.org/10.1134/S1061920819030117
Odzijewicz, A., Wawreniuk, E.: An integrable (classical and quantum) four-wave mixing Hamiltonian system. J. Math. Phys. 61(7), 073503, 18 (2020). https://doi.org/10.1063/5.0006887
Odzijewicz, A., Wawreniuk, E.: Integrable Hamiltonian systems on the symplectic realizations of e(3)∗. Russ. J. Math. Phys. 29(1), 91–114 (2022). https://doi.org/10.1134/S1061920822010095
Cannas da Silva, A., Weinstein, A.: Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, vol. 10. American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA (1999)
Walls, D.F., Milburn, G.J.: Quantum optics, second edn. Springer-Verlag, Berlin (2008). https://doi.org/10.1007/978-3-540-28574-8
Zhukovsky, N.E.: On the motion of a rigid body having cavities filled with homogeneous liquid. Zh. Russk. Fiz-Khim. Obsch. 17(6), 81–113 (1885)
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Wawreniuk, E. (2023). Symplectic Realizations of e(3)∗. In: Kielanowski, P., Dobrogowska, A., Goldin, G.A., Goliński, T. (eds) Geometric Methods in Physics XXXIX. WGMP 2022. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-30284-8_23
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DOI: https://doi.org/10.1007/978-3-031-30284-8_23
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