# American Institute of Mathematical Sciences

November  2019, 12(7): 2127-2141. doi: 10.3934/dcdss.2019137

## On periodic solutions in the Whitney's inverted pendulum problem

 Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland

Received  November 2017 Revised  April 2018 Published  December 2018

Fund Project: This research is partially supported by the Polish National Science Center under Grant No. 2014/14/A/ST1/00453.

In the book "What is Mathematics?" Richard Courant and Herbert Robbins presented a solution of a Whitney's problem of an inverted pendulum on a railway carriage moving on a straight line. Since the appearance of the book in 1941 the solution was contested by several distinguished mathematicians. The first formal proof based on the idea of Courant and Robbins was published by Ivan Polekhin in 2014. Polekhin also proved a theorem on the existence of a periodic solution of the problem provided the movement of the carriage on the line is periodic. In the present paper we slightly improve the Polekhin's theorem by lowering the regularity class of the motion and we prove a theorem on the existence of a periodic solution if the carriage moves periodically on the plane.

Citation: Roman Srzednicki. On periodic solutions in the Whitney's inverted pendulum problem. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2127-2141. doi: 10.3934/dcdss.2019137
##### References:
 [1] V. I. Arnol'd, What is Mathematics?, (Russian), MCNMO, Moscow, 2002. Google Scholar [2] V. I. Arnol'd, Mathematical Understanding of Nature, (Russian), MCNMO, Moscow, 2009. English translation: V. I. Arnold, Mathematical Understanding of Nature, Amer. Math. Soc., Providence, R. I., 2014. doi: 10.1090/mbk/085.  Google Scholar [3] B. E. Blank, Book review "What is Mathematics? An Elementary Approach to Ideas and Methods", Notices Amer. Math. Soc., 48 (2001), 1325-1329.   Google Scholar [4] S. V. Bolotin and V. V. Kozlov, Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney's inverted pendulum problem,, Izv. Math., 79 (2015), 894-901.  doi: 10.4213/im8413.  Google Scholar [5] A. Broman, A mechanical problem by H. Whitney, Nordisk Matematisk Tidskrift, 6 (1958), 78-82.   Google Scholar [6] A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.1090/S0002-9947-1992-1042285-7.  Google Scholar [7] L. Consolini and M. Tosques, On the existence of small periodic solutions for 2-dimensional inverted pendulum on a cart, SIAM J. Appl. Math., 68 (2007), 486-502.  doi: 10.1137/070683404.  Google Scholar [8] L. Consolini and M. Tosques, On the exact tracking of the spherical inverted pendulum via a homotopy method, Systems Control Lett., 58 (2009), 1-6.  doi: 10.1016/j.sysconle.2008.06.010.  Google Scholar [9] L. Consolini and M. Tosques, A continuation theorem on periodic solutions of regular nonlinear systems and its application to the exact tracking problem for the inverted spherical pendulum,, Nonlinear Anal., 74 (2011), 9-26.  doi: 10.1016/j.na.2010.08.002.  Google Scholar [10] R. Courant and H. Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, 1941.   Google Scholar [11] R. Courant and H. Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods, 2nd edition revised by Ⅰ. Stewart, Oxford University Press, 1996.  Google Scholar [12] C. Davis, Christopher Zeeman Medal Award lecture, The London Math. Soc. Newsletter, 384 (2009), 33-34.   Google Scholar [13] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [14] A. Dold, Lectures on Algebraic Topology, 2nd edition, Springer-Verlag, Berlin, Heidelberg and New York, 1980.  Google Scholar [15] R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math., 568, Springer-Verlag, Berlin, Heidelberg and New York, 1977.  Google Scholar [16] L. Gillman, Review of "What is Mathematics?" by Richard Courant and Herbert Robbins, revised by Ian Stewart, Amer. Math. Monthly, 105 (1998), 485-488.  doi: 10.2307/3109832.  Google Scholar [17] J. E. Littlewood, A Mathematician's Miscellany, Methuen & Co., London, 1953.  Google Scholar [18] J. R. Newman (ed.), The World of Mathematics, Vol. Ⅳ., Allen & Unwin, London, 1960. Google Scholar [19] I. Polekhin, Examples of topological approach to the problem of inverted pendulum with moving pivot point, (Russian), Nelin. Dinam., 10 (2014), 465-472.   Google Scholar [20] I. Polekhin, Periodic and falling-free motion of an inverted spherical pendulum with a moving pivot point, preprint, arXiv: 1411.1585. Google Scholar [21] T. Poston, Au courant with differential equations, Manifold, 18 (1976), 6-9.   Google Scholar [22] R. Srzednicki, On rest points of dynamical systems, Fund. Math., 126 (1985), 69-81.  doi: 10.4064/fm-126-1-69-81.  Google Scholar [23] R. Srzednicki, Periodic and constant solutions via topological principle of Ważewski, Univ. Iagel. Acta Math., 26 (1987), 183-190.   Google Scholar [24] R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal., 22 (1994), 707-737.  doi: 10.1016/0362-546X(94)90223-2.  Google Scholar [25] R. Srzednicki, Ważewski method and Conley index, in Handbook of Ordinary Differential Equations, Vol Ⅰ., (eds. A. Cañada, P. Drábek and A. Fonda), Elsevier/North Holland, Amsterdam, (2004), 591-684.  Google Scholar [26] I. Stewart, Gem, Set and Math, Blackwell Ltd., London, 1989.  Google Scholar [27] F. Zanolin, Bound sets, periodic solutions and flow-invariance for ordinary differential equations in $\mathbb R^n$: Some remarks, Rend. Ist. Mat. Univ. Trieste, 19 (1987), 76-92.   Google Scholar [28] O. Zubelevich, Bounded solutions to the system of second order ODEs and the Whitney pendulum, Appl. Math. (Warsaw), 42 (2015), 159-165.  doi: 10.4064/am42-2-3.  Google Scholar

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##### References:
 [1] V. I. Arnol'd, What is Mathematics?, (Russian), MCNMO, Moscow, 2002. Google Scholar [2] V. I. Arnol'd, Mathematical Understanding of Nature, (Russian), MCNMO, Moscow, 2009. English translation: V. I. Arnold, Mathematical Understanding of Nature, Amer. Math. Soc., Providence, R. I., 2014. doi: 10.1090/mbk/085.  Google Scholar [3] B. E. Blank, Book review "What is Mathematics? An Elementary Approach to Ideas and Methods", Notices Amer. Math. Soc., 48 (2001), 1325-1329.   Google Scholar [4] S. V. Bolotin and V. V. Kozlov, Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney's inverted pendulum problem,, Izv. Math., 79 (2015), 894-901.  doi: 10.4213/im8413.  Google Scholar [5] A. Broman, A mechanical problem by H. Whitney, Nordisk Matematisk Tidskrift, 6 (1958), 78-82.   Google Scholar [6] A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.1090/S0002-9947-1992-1042285-7.  Google Scholar [7] L. Consolini and M. Tosques, On the existence of small periodic solutions for 2-dimensional inverted pendulum on a cart, SIAM J. Appl. Math., 68 (2007), 486-502.  doi: 10.1137/070683404.  Google Scholar [8] L. Consolini and M. Tosques, On the exact tracking of the spherical inverted pendulum via a homotopy method, Systems Control Lett., 58 (2009), 1-6.  doi: 10.1016/j.sysconle.2008.06.010.  Google Scholar [9] L. Consolini and M. Tosques, A continuation theorem on periodic solutions of regular nonlinear systems and its application to the exact tracking problem for the inverted spherical pendulum,, Nonlinear Anal., 74 (2011), 9-26.  doi: 10.1016/j.na.2010.08.002.  Google Scholar [10] R. Courant and H. Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, 1941.   Google Scholar [11] R. Courant and H. Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods, 2nd edition revised by Ⅰ. Stewart, Oxford University Press, 1996.  Google Scholar [12] C. Davis, Christopher Zeeman Medal Award lecture, The London Math. Soc. Newsletter, 384 (2009), 33-34.   Google Scholar [13] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [14] A. Dold, Lectures on Algebraic Topology, 2nd edition, Springer-Verlag, Berlin, Heidelberg and New York, 1980.  Google Scholar [15] R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math., 568, Springer-Verlag, Berlin, Heidelberg and New York, 1977.  Google Scholar [16] L. Gillman, Review of "What is Mathematics?" by Richard Courant and Herbert Robbins, revised by Ian Stewart, Amer. Math. Monthly, 105 (1998), 485-488.  doi: 10.2307/3109832.  Google Scholar [17] J. E. Littlewood, A Mathematician's Miscellany, Methuen & Co., London, 1953.  Google Scholar [18] J. R. Newman (ed.), The World of Mathematics, Vol. Ⅳ., Allen & Unwin, London, 1960. Google Scholar [19] I. Polekhin, Examples of topological approach to the problem of inverted pendulum with moving pivot point, (Russian), Nelin. Dinam., 10 (2014), 465-472.   Google Scholar [20] I. Polekhin, Periodic and falling-free motion of an inverted spherical pendulum with a moving pivot point, preprint, arXiv: 1411.1585. Google Scholar [21] T. Poston, Au courant with differential equations, Manifold, 18 (1976), 6-9.   Google Scholar [22] R. Srzednicki, On rest points of dynamical systems, Fund. Math., 126 (1985), 69-81.  doi: 10.4064/fm-126-1-69-81.  Google Scholar [23] R. Srzednicki, Periodic and constant solutions via topological principle of Ważewski, Univ. Iagel. Acta Math., 26 (1987), 183-190.   Google Scholar [24] R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal., 22 (1994), 707-737.  doi: 10.1016/0362-546X(94)90223-2.  Google Scholar [25] R. Srzednicki, Ważewski method and Conley index, in Handbook of Ordinary Differential Equations, Vol Ⅰ., (eds. A. Cañada, P. Drábek and A. Fonda), Elsevier/North Holland, Amsterdam, (2004), 591-684.  Google Scholar [26] I. Stewart, Gem, Set and Math, Blackwell Ltd., London, 1989.  Google Scholar [27] F. Zanolin, Bound sets, periodic solutions and flow-invariance for ordinary differential equations in $\mathbb R^n$: Some remarks, Rend. Ist. Mat. Univ. Trieste, 19 (1987), 76-92.   Google Scholar [28] O. Zubelevich, Bounded solutions to the system of second order ODEs and the Whitney pendulum, Appl. Math. (Warsaw), 42 (2015), 159-165.  doi: 10.4064/am42-2-3.  Google Scholar
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