Quadratic Hamilton-Poisson systems in three dimensions: equivalence, stability, and integration. (English) Zbl 1406.70024

In this paper the authors consider Hamilton-Poisson quadratic systems on three-dimensional Lie-Poisson spaces. Positive (semi-definite) quadratic Hamilton-Poisson systems in Lee-Poisson spaces arise when invariant optimal control problems are studied on Lie groups [A. A. Agrachev and Y. L. Sachkov, Control theory from the geometric viewpoint. Berlin: Springer (2004; Zbl 1062.93001), V. Jurdjevic, Geometric control theory. Cambridge: Cambridge Univ. Press (1997; Zbl 0940.93005), R. Biggs and C. C. Remsing, Mediterr. J. Math. 11, No. 1, 193–215 (2014; Zbl 1305.93007)].
Only homogeneous (positive) semidefinite systems are investigated in this article. First, the systems are classified: an exhaustive and nonredundant list of of 23 normal forms is presented.
The classification is done in two steps: the authors classified the systems on each three-dimensional Lie-Poisson space and subsequently, equivalences of systems on non-isomorphic Lie-Poisson spaces (- on each three-dimensional Lie-Poisson space) is consider. Second, for each normal form, the stability of the equilibria is investigated and explicit expressions for the integral curves of all but three families of systems are obtained.


70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
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[1] Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Benjamin-Cummings, Reading (1978) · Zbl 0393.70001
[2] Adams, R.M., Biggs, R., Remsing, C.C.: Single-input control systems on the Euclidean group SE \((2)\mathsf{SE}(2)\). Eur. J. Pure Appl. Math. 5(1), 1-15 (2012) · Zbl 1389.49011
[3] Adams, R.M., Biggs, R., Remsing, C.C.: On some quadratic Hamilton-Poisson systems. Appl. Sci. 15, 1-12 (2013) · Zbl 1308.53122
[4] Adams, R.M., Biggs, R., Remsing, C.C.: Two-input control systems on the Euclidean group SE \((2)\mathsf{SE}(2)\). ESAIM Control Optim. Calc. Var. 19(4), 947-975 (2013). doi:10.1051/cocv/2012040 · Zbl 1283.49003
[5] Adams, R. M.; Biggs, R.; Remsing, C. C.; Mladenov, I. M. (ed.); Ludu, A. (ed.); Yoshioka, A. (ed.), Quadratic Hamilton-Poisson systems on so−∗\((3)\mathfrak{so}^*_-(3)\): classifications and integration, Varna, Bulgaria, 2013, Sophia
[6] Adams, R.M., Biggs, R., Holderbaum, W., Remsing, C.C.: On the stability and integration of Hamilton-Poisson systems on so(3)−∗\( \mathfrak{so}(3)^*_-\). Rend. Mat. Appl. In press · Zbl 1416.17022
[7] Agrachev, A.A., Sachkov, Y.L.: Control Theory from the Geometric Viewpoint. Springer, Berlin (2004) · Zbl 1062.93001
[8] Armitage, J.V., Eberlein, W.F.: Elliptic Functions. Cambridge University Press, Cambridge (2006). doi:10.1017/CBO9780511617867 · Zbl 1105.14001
[9] Aron, A.; Pop, C., Quadratic and homogeneous Hamilton-Poisson systems on the 13th Lie algebra from Bianchi’s classification, Bucharest, Romania, 2009, Bucharest · Zbl 1216.53072
[10] Aron, A., Dăniasă, C., Puta, M.: Quadratic and homogeneous Hamilton-Poisson system on (so \((3))(\text{so}(3))\). Int. J. Geom. Methods Mod. Phys. 4(7), 1173-1186 (2007). doi:10.1142/S0219887807002491 · Zbl 1171.37326
[11] Aron, A., Pop, C., Puta, M.: Some remarks on (sl(2,R))\((\text{sl}(2,\mathbb{R}))\) and Kahan’s integrator. An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. 53(suppl. 1), 49-60 (2007) · Zbl 1199.53171
[12] Aron, A., Craioveanu, M., Pop, C., Puta, M.: Quadratic and homogeneous Hamilton-Poisson systems on A3,6,−1∗\(A_{3,6,-1}^*\). Balk. J. Geom. Appl. 15(1), 1-7 (2010) · Zbl 1222.37048
[13] Barrett, D.I., Biggs, R., Remsing, C.C.: Optimal control of drift-free invariant control systems on the group of motions of the Minkowski plane. In: Proceedings of the 13th European Control Conference, Strasbourg, France, 2014, pp. 2466-2471. European Control Association (2014)
[14] Barrett, D.I., Biggs, R., Remsing, C.C.: Quadratic Hamilton-Poisson systems on se(1,1)−∗\( \mathfrak{se}(1,1)^*_-\): the homogeneous case. Int. J. Geom. Methods Mod. Phys. 12(1), 1550011, 17 pp. (2015). doi:10.1142/S0219887815500115 · Zbl 1307.37034
[15] Barrett, D.I., Biggs, R., Remsing, C.C.: Quadratic Hamilton-Poisson systems on se(1,1)−∗\( \mathfrak{se} (1,1)^*_-\): the inhomogeneous case. Preprint · Zbl 1411.53069
[16] Bartlett, C.E., Biggs, R., Remsing, C.C.: A few remarks on quadratic Hamilton-Poisson systems on the Heisenberg Lie-Poisson space. Acta Math. Univ. Comen. In press · Zbl 1374.70036
[17] Biggs, J., Holderbaum, W.: Integrable quadratic Hamiltonians on the Euclidean group of motions. J. Dyn. Control Syst. 16(3), 301-317 (2010). doi:10.1007/s10883-010-9094-8 · Zbl 1203.70041
[18] Biggs, R., Remsing, C.C.: On the equivalence of cost-extended control systems on Lie groups. In: Karimi, H.R. (ed.) Recent Researches in Automatic Control, Systems Science and Communications, Porto, Portugal, 2012, pp. 60-65. WSEAS Press (2012)
[19] Biggs, R., Remsing, C.: Cost-extended control systems on Lie groups. Mediterr. J. Math. 11(1), 193-215 (2014) · Zbl 1305.93007
[20] Biggs, R.; Remsing, C. C.; Mladenov, I. M. (ed.); Ludu, A. (ed.); Yoshioka, A. (ed.), A classification of quadratic Hamilton-Poisson systems in three dimensions, Varna, Bulgaria, 2013, Sophia · Zbl 1349.37053
[21] Biggs, R., Remsing, C.C.: Equivalence of control systems on the pseudo-orthogonal group SO \((2,1)0\mathsf{SO}(2,1)_0\). An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 24(2), 45-65 (2016). doi:10.1515/auom-2016-0027. http://www.anstuocmath.ro/mathematics//ANALE2016VOL2/Biggs_R.__Remsing_C.C..pdf · Zbl 1389.93067
[22] Craioveanu, M.; Pop, C.; Aron, A.; Petrişor, C., An optimal control problem on the special Euclidean group SE(3,R)\({\mathsf{SE}}(3, \mathbb{R})\), Bucharest, Romania, 2009, Bucharest · Zbl 1227.53088
[23] Dăniasă, C., Gîrban, A., Tudoran, R.M.: New aspects on the geometry and dynamics of quadratic Hamiltonian systems on (so \((3))(\mathfrak{so}(3))\). Int. J. Geom. Methods Mod. Phys. 8(8), 1695-1721 (2011). doi:10.1142/S0219887811005889 · Zbl 1326.70027
[24] Harvey, A.: Automorphisms of the Bianchi model Lie groups. J. Math. Phys. 20(2), 251-253 (1979). doi:10.1063/1.524073
[25] Holm, D.D.: Geometric Mechanics I: Dynamics and Symmetry. Imperial College Press, London (2008) · Zbl 1160.70001
[26] Holm, D.D.: Geometric Mechanics II: Rotating, Translating and Rolling. Imperial College Press, London (2008) · Zbl 1149.70001
[27] Holm, D.D., Marsden, J.E., Ratiu, T., Weinstein, A.: Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123(1-2), 1-116 (1985). doi:10.1016/0370-1573(85)90028-6 · Zbl 0717.76051
[28] Jurdjevic, V.: Geometric Control Theory. Cambridge University Press, Cambridge (1997) · Zbl 0940.93005
[29] Krasiński, A., Behr, C.G., Schücking, E., Estabrook, F.B., Wahlquist, H.D., Ellis, G.F.R., Jantzen, R., Kundt, W.: The Bianchi classification in the Schücking-Behr approach. Gen. Relativ. Gravit. 35(3), 475-489 (2003). doi:10.1023/A:1022382202778 · Zbl 1016.83004
[30] Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P.: Poisson Structures. Springer, Heidelberg (2013). doi:10.1007/978-3-642-31090-4 · Zbl 1284.53001
[31] Libermann, P., Marle, C.M.: Symplectic Geometry and Analytical Mechanics. Reidel, Dordrecht (1987). doi:10.1007/978-94-009-3807-6 · Zbl 0643.53002
[32] MacCallum, M. A.H., On the classification of the real four-dimensional Lie algebras, New York, 1996, New York · Zbl 0959.17003
[33] Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Springer, New York (1999) · Zbl 0933.70003
[34] Monroy-Pérez, F., Anzaldo-Meneses, A.: Optimal control on the Heisenberg group. J. Dyn. Control Syst. 5(4), 473-499 (1999). doi:10.1023/A:1021787121457 · Zbl 0956.49013
[35] Mubarakzjanov, G.M.: On solvable Lie algebras. Izv. Vysš. Učebn. Zaved., Mat. 1963(no 1 (32)), 114-123 (1963) · Zbl 0166.04104
[36] Ortega, J.P., Planas-Bielsa, V., Ratiu, T.S.: Asymptotic and Lyapunov stability of constrained and Poisson equilibria. J. Differ. Equ. 214(1), 92-127 (2005). doi:10.1016/j.jde.2004.09.016 · Zbl 1066.37017
[37] Patera, J., Sharp, R.T., Winternitz, P., Zassenhaus, H.: Invariants of real low dimension Lie algebras. J. Math. Phys. 17(6), 986-994 (1976) · Zbl 0357.17004
[38] Patrick, G.W., Roberts, M., Wulff, C.: Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods. Arch. Ration. Mech. Anal. 174(3), 301-344 (2004). doi:10.1007/s00205-004-0322-9 · Zbl 1145.37323
[39] Popovych, R.O., Boyko, V.M., Nesterenko, M.O., Lutfullin, M.W.: Realizations of real low-dimensional Lie algebras. J. Phys. A 36(26), 7337-7360 (2003). doi:10.1088/0305-4470/36/26/309 · Zbl 1040.17021
[40] Puta, M.: Hamiltonian Mechanical Systems and Geometric Quantization. Kluwer Academic, Dordrecht (1993). doi:10.1007/978-94-011-1992-4 · Zbl 0795.70001
[41] Puta, M.; Caşu, I.; Mladenov, I. M. (ed.); Naber, G. L. (ed.), Geometrical aspects in the rigid body dynamics with three quadratic controls, Varna, Bulgaria, 1999, Sofia · Zbl 0986.70009
[42] Puta, M.; Butur, M.; Goldenthal, G.; Moş, I.; Rujescu, C.; Mladenov, I. M. (ed.); Naber, G. L. (ed.), Maxwell-Bloch equations with a quadratic control about Ox \(1Ox_1\) axis, Varna, Bulgaria, 2000, Sofia · Zbl 1085.34536
[43] Remsing, C.C.: Optimal control and integrability on Lie groups. An. Univ. Vest. Timiş., Ser. Mat.-Inform. 49(2), 101-118 (2011) · Zbl 1274.49003
[44] Sachkov, Y.L.: Maxwell strata in the Euler elastic problem. J. Dyn. Control Syst. 14(2), 169-234 (2008). doi:10.1007/s10883-008-9039-7 · Zbl 1203.49004
[45] Testa, F.J.: Congruence diagonalization and Lie groups. SIAM J. Appl. Math. 28, 54-59 (1975) · Zbl 0312.15014
[46] Tudoran, R.M.: On a class of three-dimensional quadratic Hamiltonian systems. Appl. Math. Lett. 25(9), 1214-1216 (2012). doi:10.1016/j.aml.2012.02.048 · Zbl 1246.37079
[47] Tudoran, R.M.: The free rigid body dynamics: generalized versus classic. J. Math. Phys. 54(7), 072704, 10 (2013). doi:10.1063/1.4816550 · Zbl 1302.70010
[48] Tudoran, R.M., Tudoran, R.A.: On a large class of three-dimensional Hamiltonian systems. J. Math. Phys. 50(1), 012703, 9 (2009). doi:10.1063/1.3068405 · Zbl 1189.37062
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