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The geodesic flow of a sub-Riemannian metric on a solvable Lie group

Abstract

We consider the sub-Riemannian problem on the three-dimensional solvable Lie group SOLV+. The problem is based on constructing a Hamiltonian structure for a given metric by the Pontryagin Maximum Principle.

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References

  1. 1.

    A. Agrachev and D. Barilari, “Sub-Riemannian structures on 3D Lie goups,”, J. Dynam. Control Systems 18(1), 21–44 (2012).

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    A. A. Agrachev and Yu. L. Sachkov, Geometric Control Theory (Fizmatlit, Moscow, 2004) [in Russian].

    Book  Google Scholar 

  3. 3.

    E. P. Aksenov, Special Functions in Celestial Mechanics (Nauka, Moscow, 1986) [in Russian].

    MATH  Google Scholar 

  4. 4.

    U. BoscainU. and F. Rossi, “Invarihant Carnot-Carathéodory Metrics on S 3,SO(3), SL(2), and lens spaces,” SIAM J. Control and Optimization 47(4), 1851–1878 (2008).

    MathSciNet  Article  Google Scholar 

  5. 5.

    O. Calin, D. -Ch. Chang and I. Markina, “Sub-Riemannian geometry on the sphere S 3,” Canadian J. Math 61(4), 821–839 (2009).

    MathSciNet  Article  Google Scholar 

  6. 6.

    I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series, and Products, (Fizmatgiz, Moscow, 1963) [Academic Press, New York-London-Toronto, 1980).

    Google Scholar 

  7. 7.

    A. D. Mazhitova, “Sub-Riemannian problem on a three-dimensional solvable Lie group,” Vestnik KazNU. Ser. Math. Mech. Inform. (2 (65)), 11–18 (2010).

    Google Scholar 

  8. 8.

    A. D. Mazhitova, “Sub-Riemannian geodesics on the three-dimensional solvable non-nilpotent Lie group SOLV,” J. Dynam. Control Sistems 17(3), 309–322 (2012).

    MathSciNet  Article  Google Scholar 

  9. 9.

    I. Moiseev and Yu. Sachkov, “Maxwell strata in sub-Riemannian problem on the group of motions of a plane,” ESAIM Control Optim. Calc. Var. 16(2), 380–399 (2010).

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Yu. L. Sachkov, Controllability and Symmetry of Invariant Systems on Lie Groups and Homogeneous Spaces, (Fizmatlit, Moscow, 2007) [in Russian].

    MATH  Google Scholar 

  11. 11.

    Yu. Sachkov, “Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane,” ESAIM Control Optim. Calc. Var. 16(4), 1018–1039 (2010).

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    I. A. Taimanov, “Integrable geodesic flows of nonholonomic metrics,” J. Dynam. Control Systems 3(1), 129–147 (1997).

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to A. D. Mazhitova.

Additional information

Original Russian Text © A. D. Mazhitova, 2012, published in Matematicheskie Trudy, 2012, Vol. 15, No. 1, pp. 120–128.

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Mazhitova, A.D. The geodesic flow of a sub-Riemannian metric on a solvable Lie group. Sib. Adv. Math. 23, 99–105 (2013). https://doi.org/10.3103/S1055134413020041

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Keywords

  • sub-Riemannian geometry
  • left-invariant metric
  • Hamiltonian
  • geodesics