Differential and Integral Equations

On the relativistic pendulum-type equation

Antonio Ambrosetti and David Arcoya

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In the first part of this paper, we consider the equation $$ \Big ( \frac{u'}{\sqrt{1-u'^2}} \Big )'+F'(u)=0 $$ modeling, if $F'(u)=\sin u$, the motion of the free relativistic planar pendulum. Using critical point theory for non-smooth functionals, we prove the existence of non-trivial $T$ periodic solutions provided $T$ is sufficiently large.

In the second part, we show the existence of periodic solutions to the free and forced relativistic spherical pendulum, where $F'$ is substituted by $$ F'(u)-h^2\, G'(u)\sim \sin u -h^2 \frac {\cos u}{\sin^3u} , \ \ \ h\in \mathbb R . $$

Article information

Differential Integral Equations, Volume 33, Number 3/4 (2020), 91-112.

First available in Project Euclid: 21 March 2020

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Primary: 34B15: Nonlinear boundary value problems 49J40: Variational methods including variational inequalities [See also 47J20] 35Q75: PDEs in connection with relativity and gravitational theory


Ambrosetti, Antonio; Arcoya, David. On the relativistic pendulum-type equation. Differential Integral Equations 33 (2020), no. 3/4, 91--112. https://projecteuclid.org/euclid.die/1584756014

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