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 $$ F'(u)-h^2\, G'(u)\sim \sin u -h^2 \frac {\cos u}{\sin^3u} , \ \ \ h\in \mathbb R . $$

中文翻译：

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相对论摆型方程

在本文的第一部分，我们考虑方程$$ \ Big（\ frac {u'} {\ sqrt {1-u'^ 2}} \ Big）'+ F'（u）= 0 $$建模，如果$ F'（u）= \ sin u $，则是自由相对论平面摆的运动。使用非光滑泛函的临界点理论，我们证明了只要$ T $足够大，就可以存在非平凡的$ T $周期解。在第二部分中，我们显示了自由和强迫相对论球形摆的周期解的存在，其中$ F'$被$$ F'（u）-h ^ 2 \，G'（u）\ sim \代替sin u -h ^ 2 \ frac {\ cos u} {\ sin ^ 3u}，\\\ h \ in \ mathbbR。$$