# American Institute of Mathematical Sciences

September  2013, 3(3): 347-374. doi: 10.3934/mcrf.2013.3.347

## Affine-quadratic problems on Lie groups

 1 Department of Mathematics, University of Toronto, 40 St. George st, Toronto, Canada

Received  December 2012 Revised  March 2013 Published  September 2013

This paper focuses on a class of left invariant variational problems on a Lie group$\ G\$whose Lie algebra $\mathfrak{g}$ admits Cartan decomposition $\mathfrak{g}=\mathfrak{p}+\mathfrak{k}$ with the usual Lie algebraic conditions \begin{equation*} \lbrack \mathfrak{p},\mathfrak{p]\subseteq k\ },\ \mathfrak{[p},\mathfrak{ k]\subseteq p},\mathfrak{\ [k},\mathfrak{k]\subseteq k.} \end{equation*}         The Maximum Principle of optimal control leads to the Hamiltonians $H$ on $\mathfrak{g\ }$that admit spectral parameter representations with important contributions to the theory of integrable Hamiltonian systems. Particular cases will be singled out that provides natural explanations for the classical results of Fock and Moser linking Kepler's problem to the geodesics on spaces of constant curvature, C.L. Jacobi's geodesic problem on an ellipsoid and J.Moser's work on integrability based on isospectral methods. The paper also shows the relevance of this class of Hamiltonians to the elastic curves on spaces of constant curvature.
Citation: Velimir Jurdjevic. Affine-quadratic problems on Lie groups. Mathematical Control & Related Fields, 2013, 3 (3) : 347-374. doi: 10.3934/mcrf.2013.3.347
##### References:
 [1] A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Point of View," Encyclopedia of Mathematical Sciences, 87. Control Theory and Optimization, II. Springer-Verlag, Berlin, 2004.  Google Scholar [2] D. V. Anosov, A note on the Kepler problem, Jour. Dynamical and Control Syst., 8 (2002), 413-442. doi: 10.1023/A:1016386605889.  Google Scholar [3] V. Arnold, "Les Méthodes Mathématiques de la Mécanique Classique," Traduction francaise, Editions Mir, 1974. Google Scholar [4] J. Cheeger and D. Ebin, "Comparison Theorems in Riemannian Geometry," Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008.  Google Scholar [5] A. V. Bolsinov, A criterion for the completeness of a family of functions in involution that is constructed by the argument translation method, (Russian) Dokl. Akad. Nauk SSSR, 301 (1988), 1037-1040; translation in Soviet Math. Dokl., 38 (1989), 161-165.  Google Scholar [6] A. V. Bolsinov and B. Jovanović, Complete involutive algebras of functions on cotangent bundles of homogeneous spaces, Math. Zeit., 246 (2004), 213-236. doi: 10.1007/s00209-003-0596-x.  Google Scholar [7] B. Bonnard, V. Jurdjevic, I. K. Kupka and G. Sallet, Transitivity of families of invariant vector fields on semi-direct products of lie groups, Trans. Amer. Math. Soc., 271 (1982), 525-535. doi: 10.1090/S0002-9947-1982-0654849-4.  Google Scholar [8] P. Eberlein, "Geometry of Nonpositively Curved Manifolds," Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1996.  Google Scholar [9] S. Helgason, "Differential Geometry, Lie Groups, and Symmetric Spaces," Pure and Applied Mathematics, 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.  Google Scholar [10] Y. N. Fedorov and B. Jovanovic, Geodesic flows and newmann systems on steifel varieties,, Geometry and Integrability, ().   Google Scholar [11] V. A. Fock, The hydrogen atom and non-euclidean geometry, Izv. Akad. Nauk SSSR, Ser Fizika 8, 1935. Google Scholar [12] B. S. Goh, Necessary conditions for singular extremals involving multiple control variables, SIAM J. Control, 4 (1966), 716-731. doi: 10.1137/0304052.  Google Scholar [13] P. Griffiths, "Exterior Differential Calculus and the Calculus of Variations," Birkhauser, Boston, 1992. Google Scholar [14] C. G. J. Jacobi, "Vorlesungen Uber Dynamic," Druck und Verlag von G. Reimer, Berlin, 1884. Google Scholar [15] V. Jurdjevic, "Geometric Control Theory," Cambridge Studies in Advanced Mathematics 52, Cambridge University Press, 1997.  Google Scholar [16] V. Jurdjevic, Hamiltonian point of view of non-euclidean geometry and elliptic functions, System and Control Letters, Lie theory and its applications in control (Würzburg, 1999), 43 (2005), 25-41. doi: 10.1016/S0167-6911(01)00093-7.  Google Scholar [17] V. Jurdjevic, Optimal control, geometry and mechanics, Mathematical Control Theory, 227-267, Springer, New York, 1999.  Google Scholar [18] V. Jurdjevic and F. Monroy-Perez, Hamiltonian systems on Lie groups: Elastic curves, tops and constrained geodesic problems, in "Non-Linear Control Theory and its Applications," World Scientific Publishing Co., Singapore 2002, 3-52. doi: 10.1142/9789812778079_0001.  Google Scholar [19] V. Jurdjevic, Optimal control on Lie groups and integrable hamiltonian systems, Regular and Chaotic Dyn., 16 (2011), 514-535. doi: 10.1134/S156035471105008X.  Google Scholar [20] V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups, Mem. Amer. Math. Soc., 178 (2005), viii+133 pp. doi: 10.1090/memo/0838.  Google Scholar [21] V. Jurdjevic, The elliptic geodesic problem on the sphere,, in preparation., ().   Google Scholar [22] H. Knorrer, Geodesics on quadrics and a mechanical problem of C. Newmann, J. Reine Angew. Math., 334 (1982), 69-78. doi: 10.1515/crll.1982.334.69.  Google Scholar [23] J. Langer and D. Singer, Knotted elastic curves in $\mathbbR ^3$, J. London. Math. Soc., (2), 30 (1984), 512-520. doi: 10.1112/jlms/s2-30.3.512.  Google Scholar [24] P. Lee, "Kepler's Problem," Master's thesis, University of Toronto, 2004. Google Scholar [25] J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure Appl. Math., 23 (1970), 609-636. doi: 10.1002/cpa.3160230406.  Google Scholar [26] J. Moser, Geometry of quadrics and spectral theory, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), 147-188, Springer, New York-Berlin, 1980.  Google Scholar [27] J. Moser, "Integrable Hamiltonian Systems and Spectral Theory," Lezioni Fermiane. [Fermi Lectures] Scuola Normale Superiore, Pisa, 1983.  Google Scholar [28] C. Newmann, De probleme quodam mechanico, quod ad primam integralium ultra-ellipticoram classem revocatum, J. Reine Angew. Math., 56 (1856), 345-378. Google Scholar [29] Y. Osipov, The Kepler problem and geodesic flows in spaces of constant curvature, Celestial Mechanics, 16 (1977), 191-208. doi: 10.1007/BF01228600.  Google Scholar [30] A. M. Perelomov, "Integrable Systems Of Classical Mechanics And Lie Algebras, Vol. I," Translated from the Russian by A. G. Reyman [A. G. Reiman]. Birkhauser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.  Google Scholar [31] T. Ratiu, The C. Newmann problem as a completely integrable system on an adjoint orbit, Trans. Amer. Math. Soc., 264 (1981), 321-329. doi: 10.2307/1998542.  Google Scholar [32] A. G. Reyman, Integrable hamiltonian systems connected with graded Lie algebras, J. Sov. Math., 19 (1982), 1507-1545. Google Scholar [33] A. G. Reyman and S. Tian- Shansky, Group theoretic methods in the theory of finite dimensional integrable systems, Encyclopaedia of Mathematical Sciences, edited by V. I. Arnold and S. P. Novikov, Part II, Chapter 2, Springer-Verlag, 1994. doi: 10.1007/978-3-642-57884-7.  Google Scholar

show all references

##### References:
 [1] A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Point of View," Encyclopedia of Mathematical Sciences, 87. Control Theory and Optimization, II. Springer-Verlag, Berlin, 2004.  Google Scholar [2] D. V. Anosov, A note on the Kepler problem, Jour. Dynamical and Control Syst., 8 (2002), 413-442. doi: 10.1023/A:1016386605889.  Google Scholar [3] V. Arnold, "Les Méthodes Mathématiques de la Mécanique Classique," Traduction francaise, Editions Mir, 1974. Google Scholar [4] J. Cheeger and D. Ebin, "Comparison Theorems in Riemannian Geometry," Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008.  Google Scholar [5] A. V. Bolsinov, A criterion for the completeness of a family of functions in involution that is constructed by the argument translation method, (Russian) Dokl. Akad. Nauk SSSR, 301 (1988), 1037-1040; translation in Soviet Math. Dokl., 38 (1989), 161-165.  Google Scholar [6] A. V. Bolsinov and B. Jovanović, Complete involutive algebras of functions on cotangent bundles of homogeneous spaces, Math. Zeit., 246 (2004), 213-236. doi: 10.1007/s00209-003-0596-x.  Google Scholar [7] B. Bonnard, V. Jurdjevic, I. K. Kupka and G. Sallet, Transitivity of families of invariant vector fields on semi-direct products of lie groups, Trans. Amer. Math. Soc., 271 (1982), 525-535. doi: 10.1090/S0002-9947-1982-0654849-4.  Google Scholar [8] P. Eberlein, "Geometry of Nonpositively Curved Manifolds," Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1996.  Google Scholar [9] S. Helgason, "Differential Geometry, Lie Groups, and Symmetric Spaces," Pure and Applied Mathematics, 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.  Google Scholar [10] Y. N. Fedorov and B. Jovanovic, Geodesic flows and newmann systems on steifel varieties,, Geometry and Integrability, ().   Google Scholar [11] V. A. Fock, The hydrogen atom and non-euclidean geometry, Izv. Akad. Nauk SSSR, Ser Fizika 8, 1935. Google Scholar [12] B. S. Goh, Necessary conditions for singular extremals involving multiple control variables, SIAM J. Control, 4 (1966), 716-731. doi: 10.1137/0304052.  Google Scholar [13] P. Griffiths, "Exterior Differential Calculus and the Calculus of Variations," Birkhauser, Boston, 1992. Google Scholar [14] C. G. J. Jacobi, "Vorlesungen Uber Dynamic," Druck und Verlag von G. Reimer, Berlin, 1884. Google Scholar [15] V. Jurdjevic, "Geometric Control Theory," Cambridge Studies in Advanced Mathematics 52, Cambridge University Press, 1997.  Google Scholar [16] V. Jurdjevic, Hamiltonian point of view of non-euclidean geometry and elliptic functions, System and Control Letters, Lie theory and its applications in control (Würzburg, 1999), 43 (2005), 25-41. doi: 10.1016/S0167-6911(01)00093-7.  Google Scholar [17] V. Jurdjevic, Optimal control, geometry and mechanics, Mathematical Control Theory, 227-267, Springer, New York, 1999.  Google Scholar [18] V. Jurdjevic and F. Monroy-Perez, Hamiltonian systems on Lie groups: Elastic curves, tops and constrained geodesic problems, in "Non-Linear Control Theory and its Applications," World Scientific Publishing Co., Singapore 2002, 3-52. doi: 10.1142/9789812778079_0001.  Google Scholar [19] V. Jurdjevic, Optimal control on Lie groups and integrable hamiltonian systems, Regular and Chaotic Dyn., 16 (2011), 514-535. doi: 10.1134/S156035471105008X.  Google Scholar [20] V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups, Mem. Amer. Math. Soc., 178 (2005), viii+133 pp. doi: 10.1090/memo/0838.  Google Scholar [21] V. Jurdjevic, The elliptic geodesic problem on the sphere,, in preparation., ().   Google Scholar [22] H. Knorrer, Geodesics on quadrics and a mechanical problem of C. Newmann, J. Reine Angew. Math., 334 (1982), 69-78. doi: 10.1515/crll.1982.334.69.  Google Scholar [23] J. Langer and D. Singer, Knotted elastic curves in $\mathbbR ^3$, J. London. Math. Soc., (2), 30 (1984), 512-520. doi: 10.1112/jlms/s2-30.3.512.  Google Scholar [24] P. Lee, "Kepler's Problem," Master's thesis, University of Toronto, 2004. Google Scholar [25] J. Moser, Regularization of Kepler's problem and the averaging method on a manifold, Comm. Pure Appl. Math., 23 (1970), 609-636. doi: 10.1002/cpa.3160230406.  Google Scholar [26] J. Moser, Geometry of quadrics and spectral theory, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), 147-188, Springer, New York-Berlin, 1980.  Google Scholar [27] J. Moser, "Integrable Hamiltonian Systems and Spectral Theory," Lezioni Fermiane. [Fermi Lectures] Scuola Normale Superiore, Pisa, 1983.  Google Scholar [28] C. Newmann, De probleme quodam mechanico, quod ad primam integralium ultra-ellipticoram classem revocatum, J. Reine Angew. Math., 56 (1856), 345-378. Google Scholar [29] Y. Osipov, The Kepler problem and geodesic flows in spaces of constant curvature, Celestial Mechanics, 16 (1977), 191-208. doi: 10.1007/BF01228600.  Google Scholar [30] A. M. Perelomov, "Integrable Systems Of Classical Mechanics And Lie Algebras, Vol. I," Translated from the Russian by A. G. Reyman [A. G. Reiman]. Birkhauser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.  Google Scholar [31] T. Ratiu, The C. Newmann problem as a completely integrable system on an adjoint orbit, Trans. Amer. Math. Soc., 264 (1981), 321-329. doi: 10.2307/1998542.  Google Scholar [32] A. G. Reyman, Integrable hamiltonian systems connected with graded Lie algebras, J. Sov. Math., 19 (1982), 1507-1545. Google Scholar [33] A. G. Reyman and S. Tian- Shansky, Group theoretic methods in the theory of finite dimensional integrable systems, Encyclopaedia of Mathematical Sciences, edited by V. I. Arnold and S. P. Novikov, Part II, Chapter 2, Springer-Verlag, 1994. doi: 10.1007/978-3-642-57884-7.  Google Scholar
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