Abstract:
For non-autonomous Lagrangian systems we introduce the notion
of a dynamically convex domain with respect to the Lagrangian. We establish
the solubility of boundary-value problems in compact dynamically convex
domains. If the Lagrangian is time-periodic, then such a domain contains
a periodic trajectory. The proofs use the Hamilton principle and known
tools of the calculus of variations in the large. Our general results are
applied to Whitney's problem on the existence of motions of an inverted
pendulum without falls.
Citation:
S. V. Bolotin, 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., 79:5 (2015), 894–901
\Bibitem{BolKoz15}
\by S.~V.~Bolotin, V.~V.~Kozlov
\paper Calculus of variations in the large, existence of trajectories in a~domain with boundary, and Whitney's inverted pendulum problem
\jour Izv. Math.
\yr 2015
\vol 79
\issue 5
\pages 894--901
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Linking options:
https://www.mathnet.ru/eng/im8413
https://doi.org/10.1070/IM2015v079n05ABEH002765
https://www.mathnet.ru/eng/im/v79/i5/p39
This publication is cited in the following 17 articles:
D. D. Kulminskiy, M. V. Malyshev, “Experimental Study of the Accuracy of a Pendulum Clock with a Vibrating Pivot Point”, Rus. J. Nonlin. Dyn., 20:4 (2024), 553–563
E. I. Kugushev, T. V. Salnikova, “Existence of Localized Motions in the Vicinity of an Unstable Equilibrium Position”, Proc. Steklov Inst. Math., 327 (2024), 118–129
Ivan Polekhin, “Asymptotically stable non-falling solutions of the Kapitza-Whitney pendulum”, Meccanica, 2023
Ivan Yu. Polekhin, “The Spherical Kapitza – Whitney Pendulum”, Regul. Chaotic Dyn., 27:1 (2022), 65–76
I. Yu. Polekhin, “The existence proof for forced oscillations by adding dissipative forces in the example of a spherical pendulum”, Theoret. and Math. Phys., 211:2 (2022), 692–700
Nikolay Stepanov, Mikhail Skvortsov, “Inverted pendulum driven by a horizontal random force: statistics of the never-falling trajectory and supersymmetry”, SciPost Phys., 13:2 (2022)
I. Yu. Polekhin, “Remarks on Forced Oscillations in Some Systems with Gyroscopic Forces”, Rus. J. Nonlin. Dyn., 16:2 (2020), 343–353
Ivan Yu. Polekhin, “The Method of Averaging for the Kapitza – Whitney Pendulum”, Regul. Chaotic Dyn., 25:4 (2020), 401–410
Ivan Yu. Polekhin, “Some Results on the Existence of Forced Oscillations in Mechanical Systems”, Proc. Steklov Inst. Math., 310 (2020), 250–261
N. A. Stepanov, M. A. Skvortsov, “Lyapunov exponent for Whitney's problem with random drive”, JETP Letters, 112:6 (2020), 376–382
Ivan Polekhin, 2020 International Conference Nonlinearity, Information and Robotics (NIR), 2020, 1
R. Srzednicki, “On periodic solutions in the whitney's inverted pendulum problem”, Discret. Contin. Dyn. Syst.-Ser. S, 12:7 (2019), 2127–2141
I. Polekhin, “On topological obstructions to global stabilization of an inverted pendulum”, Syst. Control Lett., 113 (2018), 31–35
S. Ozana, M. Schlegel, “Computation of reference trajectories for inverted pendulum with the use of two-point BvP with free parameters”, IFAC PAPERSONLINE, 51:6 (2018), 408–413
I. Polekhin, “On motions without falling of an inverted pendulum with dry friction”, J. Geom. Mech., 10:4 (2018), 411–417
I. Yu. Polekhin, “On the impossibility of global stabilization of the Lagrange top”, Mech. Sol., 53:2 (2018), S71–S75
Polekhin I., “A Topological View on Forced Oscillations and Control of An Inverted Pendulum”, Geometric Science of Information, Gsi 2017, Lecture Notes in Computer Science, 10589, eds. Nielsen F., Barbaresco F., Springer International Publishing Ag, 2017, 329–335
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