Periodic Solutions of the Forced Relativistic Pendulum
(2010) Differential and Integral Equations — Vol. 23, n° 9-10, p. 801-810 (2010)
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- Abstract
- The existence of at least one classical T-periodic solution is proved for differential equations of the form (phi(u'))' - g(x, u) = h(x) when phi : (-a, a) -> R is an increasing homeomorphism, g is a Carathedory function T-periodic with respect to x, 2 pi-periodic with respect to u, of mean value zero with respect to u, and h is an element of L-loc(1)(R) is T-periodic and has mean value zero. The problem is reduced to finding a minimum for the corresponding action integral over a closed convex subset of the space of T-periodic Lipschitz functions, and then to show, using variational inequalities techniques, that such a minimum solves the differential equation. A special case is the "relativistic forced pendulum equation" (u'/root 1-u'(2))' + A sin u = h(x).
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Brezis, H., & Mawhin, J. (2010). Periodic Solutions of the Forced Relativistic Pendulum. Differential and Integral Equations, 23(9-10), 801-810. https://hdl.handle.net/2078.5/131426 (Original work published 2010)