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March  2021, 13(1): 55-72. doi: 10.3934/jgm.2020031

A Lagrangian approach to extremal curves on Stiefel manifolds

Knut Hüper 1,, , Irina Markina 2, and  Fátima Silva Leite 3,

1. 

Institute of Mathematics, Julius-Maximilians-Universität Würzburg, Germany
2. 

Department of Mathematics, University of Bergen, Norway
3. 

Institute of Systems and Robotics, and Department of Mathematics, University of Coimbra, Portugal

* Corresponding author

Received  March 2020 Published  March 2021 Early access  November 2020

A unified framework for studying extremal curves on real Stiefel manifolds is presented. We start with a smooth one-parameter family of pseudo-Riemannian metrics on a product of orthogonal groups acting transitively on Stiefel manifolds. In the next step Euler-Langrange equations for a whole class of extremal curves on Stiefel manifolds are derived. This includes not only geodesics with respect to different Riemannian metrics, but so-called quasi-geodesics and smooth curves of constant geodesic curvature, as well. It is shown that they all can be written in closed form. Our results are put into perspective to recent related work where a Hamiltonian rather than a Lagrangian approach was used. For some specific values of the parameter we recover certain well-known results.

Keywords: Extremal curves, geodesics, indefinite metric, Killing form, Lagrange multiplier, smooth distributions, Stiefel manifold, variational calculus.
Mathematics Subject Classification: Primary: 58E10, 53C22, 53C30.
Citation: Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2021, 13 (1) : 55-72. doi: 10.3934/jgm.2020031
References:
[1] P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2008.  doi: 10.1515/9781400830244.
[2]

R. L. Adler, J.-P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal., 22 (2002), 359-390.  doi: 10.1093/imanum/22.3.359.

[3] A. Agrachev, D. Barilari and U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge University Press, Cambridge, 2020.  doi: 10.1017/9781108677325.
[4]

A. Bloch, L. Colombo, R. Gupta and D. Martín de Diego, A geometric approach to the optimal control of nonholonomic mechanical systems, in Analysis and Geometry in Control Theory and Its Applications, Springer, Cham, 2015, 35–64. doi: 10.1007/978-3-319-06917-3_2.

[5]

A. M. Bloch, Nonholonomic Mechanics and Control, 2nd edition, Springer, New York, 2015. doi: 10.1007/978-1-4939-3017-3.

[6]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Nonholonomic dynamics, Notices Amer. Math. Soc., 52 (2005), 324-333. 

[7]

A. M. Bloch, P. E. Crouch and A. K. Sanyal, A variational problem on Stiefel manifolds, Nonlinearity, 19 (2006), 2247-2276.  doi: 10.1088/0951-7715/19/10/002.

[8]

R. W. Brockett, Finite dimensional linear systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2015. doi: 10.1137/1.9781611973884.

[9]

A. Edelman, T. A. Arias and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1998), 303-353.  doi: 10.1137/S0895479895290954.

[10]

J. Faraut, Analysis on Lie Groups. An Introduction, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511755170.

[11]

Y. N. Fedorov and B. Jovanović, Geodesic flows and Neumann systems on Stiefel varieties: Geometry and integrability, Math. Z., 270 (2012), 659-698.  doi: 10.1007/s00209-010-0818-y.

[12]

I. Gohberg, P. Lancaster and L. Rodman, Indefinite Linear Algebra and Applications, Birkhäuser Verlag, Basel, 2005. doi: 10.1007/b137517.

[13]

K. Hüper, M. Kleinsteuber and F. Silva Leite, Rolling Stiefel manifolds, Internat. J. Systems Sci., 39 (2008), 881-887.  doi: 10.1080/00207720802184717.

[14]

K. Hüper, I. Markina and F. Silva Leite, An extrinsic approach to sub-Riemannian geodesics on the orthogonal group, in CONTROLO 2020, Proceedings of the 14th APCA International Conference on Automatic Control and Soft Computing, vol. 695, LNEE, Bragança, Portugal, Springer 2020,274–283. doi: 10.1007/978-3-030-58653-9_26.

[15]

K. Hüper and F Ullrich, Real Stiefel manifolds: An extrinsic point of view, In 13th APCA International Conference on Automatic Control and Soft Computing (Controlo 2018), Ponta Delgada, Azores, Portugal, 2018, 13–18. doi: 10.1109/CONTROLO.2018.8514292.

[16]

I. I. Hussein and A. M. Bloch, Optimal control of underactuated nonholonomic mechanical systems, IEEE Trans. Automat. Control, 53 (2008), 668-682.  doi: 10.1109/TAC.2008.919853.

[17] V. Jurdjevic, Optimal Control and Geometry: Integrable Systems, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316286852.
[18]

V. Jurdjevic, I. Markina and F. Silva Leite, Extremal curves on Stiefel and Grassmann manifolds, Journal of Geometric Analysis, 30 (2020), 3948–3978. doi: 10.1007/s12220-019-00223-1.

[19]

V. Jurdjevic, F. Silva Leite and K. Krakowski, The geometry of quasi-geodesics on Stiefel manifolds, in 13th APCA International Conference on Automatic Control and Soft Computing (Controlo 2018), Ponta Delgada, Azores, Portugal, 2018,213–218. doi: 10.1109/CONTROLO.2018.8514270.

[20]

K. A. Krakowski, L. Machado, F. Silva Leite and J. Batista, A modified Casteljau algorithm to solve interpolation problems on Stiefel manifolds, J. Comput. Appl. Math., 311 (2017), 84-99.  doi: 10.1016/j.cam.2016.07.018.

[21]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society, Providence, RI, 2002. doi: 10.1090/surv/091.

[22]

Y. Nishimori and S. Akaho, Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold, Neurocomputing, 67 (2005), 106-135.  doi: 10.1016/j.neucom.2004.11.035.

[23]

A. L. Onishchik, Topology of Transitive Transformation Groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994.

[24]

J. Roe, Lectures on Coarse Geometry, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/ulect/031.

[25]

J. Roe, What is $\dots$ a coarse space?, Notices Amer. Math. Soc., 53 (2006), 668-669. 

[26]

E. Stiefel, Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten, Comment. Math. Helv., 8 (1935), 305-353. 

[27]

F. Ullrich, Rolling maps for real Stiefel manifolds, Master's thesis, Institute of Mathematics, Julius-Maximilians-Universität Würzburg, Germany, 2017.

[28]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. 3, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-5020-3.

show all references

References:
[1] P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2008.  doi: 10.1515/9781400830244.
[2]

R. L. Adler, J.-P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal., 22 (2002), 359-390.  doi: 10.1093/imanum/22.3.359.

[3] A. Agrachev, D. Barilari and U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge University Press, Cambridge, 2020.  doi: 10.1017/9781108677325.
[4]

A. Bloch, L. Colombo, R. Gupta and D. Martín de Diego, A geometric approach to the optimal control of nonholonomic mechanical systems, in Analysis and Geometry in Control Theory and Its Applications, Springer, Cham, 2015, 35–64. doi: 10.1007/978-3-319-06917-3_2.

[5]

A. M. Bloch, Nonholonomic Mechanics and Control, 2nd edition, Springer, New York, 2015. doi: 10.1007/978-1-4939-3017-3.

[6]

A. M. Bloch, J. E. Marsden and D. V. Zenkov, Nonholonomic dynamics, Notices Amer. Math. Soc., 52 (2005), 324-333. 

[7]

A. M. Bloch, P. E. Crouch and A. K. Sanyal, A variational problem on Stiefel manifolds, Nonlinearity, 19 (2006), 2247-2276.  doi: 10.1088/0951-7715/19/10/002.

[8]

R. W. Brockett, Finite dimensional linear systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2015. doi: 10.1137/1.9781611973884.

[9]

A. Edelman, T. A. Arias and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1998), 303-353.  doi: 10.1137/S0895479895290954.

[10]

J. Faraut, Analysis on Lie Groups. An Introduction, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511755170.

[11]

Y. N. Fedorov and B. Jovanović, Geodesic flows and Neumann systems on Stiefel varieties: Geometry and integrability, Math. Z., 270 (2012), 659-698.  doi: 10.1007/s00209-010-0818-y.

[12]

I. Gohberg, P. Lancaster and L. Rodman, Indefinite Linear Algebra and Applications, Birkhäuser Verlag, Basel, 2005. doi: 10.1007/b137517.

[13]

K. Hüper, M. Kleinsteuber and F. Silva Leite, Rolling Stiefel manifolds, Internat. J. Systems Sci., 39 (2008), 881-887.  doi: 10.1080/00207720802184717.

[14]

K. Hüper, I. Markina and F. Silva Leite, An extrinsic approach to sub-Riemannian geodesics on the orthogonal group, in CONTROLO 2020, Proceedings of the 14th APCA International Conference on Automatic Control and Soft Computing, vol. 695, LNEE, Bragança, Portugal, Springer 2020,274–283. doi: 10.1007/978-3-030-58653-9_26.

[15]

K. Hüper and F Ullrich, Real Stiefel manifolds: An extrinsic point of view, In 13th APCA International Conference on Automatic Control and Soft Computing (Controlo 2018), Ponta Delgada, Azores, Portugal, 2018, 13–18. doi: 10.1109/CONTROLO.2018.8514292.

[16]

I. I. Hussein and A. M. Bloch, Optimal control of underactuated nonholonomic mechanical systems, IEEE Trans. Automat. Control, 53 (2008), 668-682.  doi: 10.1109/TAC.2008.919853.

[17] V. Jurdjevic, Optimal Control and Geometry: Integrable Systems, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316286852.
[18]

V. Jurdjevic, I. Markina and F. Silva Leite, Extremal curves on Stiefel and Grassmann manifolds, Journal of Geometric Analysis, 30 (2020), 3948–3978. doi: 10.1007/s12220-019-00223-1.

[19]

V. Jurdjevic, F. Silva Leite and K. Krakowski, The geometry of quasi-geodesics on Stiefel manifolds, in 13th APCA International Conference on Automatic Control and Soft Computing (Controlo 2018), Ponta Delgada, Azores, Portugal, 2018,213–218. doi: 10.1109/CONTROLO.2018.8514270.

[20]

K. A. Krakowski, L. Machado, F. Silva Leite and J. Batista, A modified Casteljau algorithm to solve interpolation problems on Stiefel manifolds, J. Comput. Appl. Math., 311 (2017), 84-99.  doi: 10.1016/j.cam.2016.07.018.

[21]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society, Providence, RI, 2002. doi: 10.1090/surv/091.

[22]

Y. Nishimori and S. Akaho, Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold, Neurocomputing, 67 (2005), 106-135.  doi: 10.1016/j.neucom.2004.11.035.

[23]

A. L. Onishchik, Topology of Transitive Transformation Groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994.

[24]

J. Roe, Lectures on Coarse Geometry, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/ulect/031.

[25]

J. Roe, What is $\dots$ a coarse space?, Notices Amer. Math. Soc., 53 (2006), 668-669. 

[26]

E. Stiefel, Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten, Comment. Math. Helv., 8 (1935), 305-353. 

[27]

F. Ullrich, Rolling maps for real Stiefel manifolds, Master's thesis, Institute of Mathematics, Julius-Maximilians-Universität Würzburg, Germany, 2017.

[28]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. 3, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-5020-3.

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