Casimirs and Lax operators from the structure of Lie algebras

[1] N. Abazari and I. Sager. Planning rigid body motions and optimal control problem on Lie group SO(2,1). Engineering and Technology, 64:448–452, 2010. [2] J.D. Biggs and W. Holderbaum. The Geometry of Optimal Control Solutions on some Six Dimensional Lie Groups. Proceedings of the 44th IEEE Conference on Decision and Control, (2):1427–1432, 2005. [3] J.D. Biggs and N. Horri. Optimal geometric motion planning for spin-stabilized spacecraft. System and Control Letters, 2012. [4] A.M. Bloch. Nonholonomic Mechanics and Control: With the Collaboration of J.Baillieul, P.Crouch and J.Marsden (Interdisciplinary Applied Mathematics). Springer, 2003. [5] F. Bullo and A.D. Lewis. Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems (Texts in Applied Mathematics). Springer, 2005. [6] M. Craioveanu, C. Pop, A. Aron, and C. Petri. An optimal control problem on the special Euclidean group SE ( 3 , R ). In International Conference "Differential GeometryDynamical Systems 2009", pages 68–78, 2009. [7] D. D’Alessandro. Algorithms for quantum control based on decompositions of Lie groups. Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), pages 967–968, 2000. [8] B. Hall. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (Graduate Texts in Mathematics). Springer, 2004. [9] D.D. Holm. Geometric Mechanics, Part I: Dynamics and Symmetry. Imperial College Press, 2008. [10] V. Jurdjevic. Geometric Control Theory (Cambridge Studies in Advanced Mathematics). Cambridge University Press, 1997. [11] V. Jurdjevic. Integrable Hamiltonian Systems on Complex Lie Groups. Memoirs of the American Mathematical Society, 178(838), 2005. [12] E.W. Justh and P.S. Krishnaprasad. Optimal natural frames. Communications in Information and Systems, 11(1):17–34, 2011. [13] P.D. Lax. Integrals of Nonlinear Equations of Evolution and Solitary. Technical Report January, Courant Institute of Mathematical Sciences, 1968. [14] N. E. Leonard and P.S. Krishnaprasad. High Order Averaging onLie Groups and Control of an Autononmous Underwater Vehicle. Proceedings of the American Control Conference, June(1):2–7, 1994. [15] J.E. Marsden, S.T. Ratiu, F. Scheck, and M.E. Mayer. Introduction to Mechanics and Symmetry and Mechanics: From Newton’s Laws to Deterministic Chaos. American Institute of Physics, 1998. [16] C.C. Remsing. Control and Integrability on SO ( 3 ). In World Congress on Engineering, volume III, 2010. [17] A.G. Reyman and M.A. Semenov-Tian-Shansky. Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. Inventiones Mathematicae, 54(1):81–100, February 1979. [18] A.G. Reyman and M.A. Semenov-Tian-Shansky. Integrable Systems II: Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems. In Dynamical systems. VII, Encyclopaedia of Mathematical Sciences, vol. 16, volume 1, page 341. Springer, 1994. [19] O.K. Sheinman. Lax equations and knizhnik-zamolodchikov connection. aiXiv.1009.4706v2, pages 1–21, 2011.