A non-existence result for periodic solutions of the relativistic pendulum with friction

Applied Mathematics Letters 144 (2023) 108697 Contents lists available at ScienceDirect Applied Mathematics Letters www.elsevier.com/locate/aml A non-existence result for periodic solutions of the relativistic pendulum with friction Pedro J. Torres1 Departamento de Matemática Aplicada, and Research Unit “Modeling Nature” (MNat), Universidad de Granada, Spain a r t i c l e i n f o Article history: Received 12 March 2023 Received in revised form 14 April 2023 Accepted 14 April 2023 Available online 23 April 2023 Keywords: Relativistic pendulum Periodic solution Linear friction abstract It is proved that if the damped periodically forced Newtonian pendulum does not have periodic solutions, the same happens for the relativistic version of the problem for high values of the parameter crepresenting the speed of light in the vacuum. ©2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 1. Introduction and main result In this paper, we present for the first time in the literature a non-existence result for periodic solutions for the forced pendulum equation with relativistic acceleration ⎛ ⎝ x′ √1−x′2 c2 ⎞ ⎠ ′ +kx′+asin x=p(t) (1.1) where c > 0 is the speed of light in the vacuum, k≥0 is a possible viscous friction coefficient, a > 0 and p is a continuous and T-periodic forcing term with mean value p=1 T∫T 0p(t)dt = 0. The classical forced pendulum equation x′′ +kx′+asin x=p(t),(1.2) has been a fundamental source of inspiration for researchers working on Dynamical Systems for many years [1,2]. Concerning the existence of periodic solutions, the history comes back one century ago when E-mail address: [email protected]. 1Research supported by AEI, Spain project MCIU-22-PID2021-128418NA-I00. Funding for open access charge: Universidad de Granada / CBUA https://doi.org/10.1016/j.aml.2023.108697 0893-9659/©2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). P.J. Torres Applied Mathematics Letters 144 (2023) 108697 Hamel [3] in 1922 proved that there exists at least one T-periodic solution when k= 0 and p(t) = bsin t. The proof is of variational nature. The general proof for any forcing term with zero mean value was given independently by Dancer [4] and Willem [5]. A conjecture by J. Mawhin [6], asking if a topological approach may be useful to prove the existence of periodic solutions in the presence of friction, generated considerable interest in the community. By the time, many specialists expected a positive answer to Mawhin’s conjecture for a good reason: by Massera’s theorem, if there are no periodic solutions then all the solutions must be unbounded, but then the addition of a friction term would turn a periodic motion into unbounded, which seems counter-intuitive. In spite of this consideration, Ortega [7] devised a remarkable example of Eq. (1.2) without periodic solutions. A second example was constructed in [8] by using a different idea. Finally, [9] provided the most general result for non-existence in the Newtonian case. From now on, let us denote by CTthe Banach space of the continuous and T-periodic functions and by ˜ CTthe space of the functions of CTwith zero mean value. Theorem 1 ([9]).Given positive constants a, k and T, there exists p∈˜ CTsuch that Eq. (1.2) has no T-periodic solutions. In the relativistic case, this result is not true: it was proved in [10] that if 2cT ≤1, Eq. (1.2) has at least one T-periodic solution for any values a, k and for any p∈˜ CT. Other sufficient conditions for existence can be found in [11–16], but non-existence results are not available in the literature up to the date. Our main result partially fills this gap. Theorem 2. Let p∈˜ CTbe such that (1.2) has no T-periodic solutions. Then, there exists c∗>0such that (1.1) (with the same choice of k, a, p) has no T-periodic solutions for any c > c∗. The proof relies on a priori bounds of solutions not depending on cand a pass to the limit via Ascol´ı–Arzela Theorem. 2. Proof of the main result Let ∥.∥∞be the usual norm of the supremum. Any periodic solution of (1.1) has a natural bound ∥x′∥∞< c. A key point of the proof is a different priori bound for the derivative given in the next lemma. Lemma 1. Any T-periodic solution of (1.1) satisfies the bound ∥x′∥∞<∥p∥2√T+aT. (2.3) Proof . Suppose that x(t) is a given T-periodic solution. Eq. (1.1) can be written as x′′ (1−x′2 c2)3/2+kx′+asin x=p(t).(2.4) Multiplying by x′′ and integrating, a basic application of Cauchy–Schwarz inequality gives ∥x′′∥2 2<∫T 0 (x′′)2 (1−x′2 c2)2/3dt =∫T 0 (p(t)−asin x)x′′dt ≤(∥p∥2+a√T)∥x′′∥2, from where ∥x′′∥2<∥p∥2+a√T . 2 P.J. Torres Applied Mathematics Letters 144 (2023) 108697 Now, taking t0∈[0, T ) such that x′(t0) = 0, |x′(t)|=⏐⏐⏐⏐∫t t0 x′′(s)ds⏐⏐⏐⏐≤ ∥x′′∥1≤√T∥x′′∥2<∥p∥2√T+aT for any t∈[0, T ]. □ Let us prove now the main result. By a contrapositive argument, let us assume that there exists a sequence cn→+∞and corresponding T-periodic solutions xn(t)of(1.1) with c=cn. By the 2π-periodic character of the nonlinearity, it is not restrictive to assume that xn(0) ∈[−π, π]. Then, we can derive the uniform bound |xn(t)−xn(0)|=⏐⏐⏐⏐∫t 0 x′ n(s)ds⏐⏐⏐⏐≤T∥x′∥∞≤ ∥p∥2T√T+aT 2. Hence, the sequence of xn(t) and its derivatives are uniformly bounded, so Ascoli-Arzela Theorem implies that a subsequence of xn(t) (not necessarily xn(t) itself, but we keep the notation for convenience) is uniformly convergent in CT. Besides, the sequence of x′′ nare also uniformly bounded because the nonlinearity and the forcing term are bounded, therefore xnis uniformly convergent to a certain x∞(t) in C1 T, the space of T-periodic functions with continuous derivatives. Note that all the derived bounds are independent of cn. The final step is to write the equation as an integral equation and a pass to the limit. Starting from (2.4), we write x′′ −x=(1−x′2 c2)3/2 (p(t)−kx′−asin x)−x, (2.5) then to find a T-periodic solution of (1.1) is equivalent that to find a T-periodic solution of the integral equation x=∫T 0 G(t, s)[(1−x′2(s) c2)3/2 (p(s)−kx′(s)−asin x(s)) −x(s)]ds, (2.6) where G(t, s) is the Green function of the linear operator x′′ −xwith periodic conditions. Consequently, xn verifies xn=∫T 0 G(t, s)[(1−x′2 n(s) c2 n)3/2 (p(s)−kx′ n(s)−asin xn(s)) −xn(s)]ds. (2.7) The Green function G(t, s) has an explicit expression, but for our purposes is it enough to know that it is uniformly bounded on the square [0, T ]×[0, T ]. Taking limits when n→+∞, the uniform boundedness of G(t, s) implies that the limit can pass inside the integral and we get x∞=∫T 0 G(t, s) [(p(s)−kx′ ∞(s)−asin x∞(s)) −x∞(s)] ds, or equivalently, x∞(t) is a T-periodic solution of the Newtonian Eq. (1.2). This concludes the proof. Data availability No data was used for the research described in the article. Acknowledgment I am indebted to Prof. R. Ortega, who communicated to me the main idea of the proof. I am also grateful to an anonymous referee for pointing out some inaccuracies. 3 P.J. Torres Applied Mathematics Letters 144 (2023) 108697 References [1] J. Mawhin, The forced pendulum equation: a paradigm fo rnonlinear analysis and dynamical systems, Expo. Math. 6 (1988) 271–287. [2] J. Mawhin, Seventy-five years of global analysis around the forced pendulum equation, Proc. 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