The motion of systems with linear restoring forces and recurring nonlinear perturbations is of central importance in physics. When a system’s natural oscillation frequencies and the frequency of the nonlinear restoring forces satisfy certain algebraic relations, the dynamics become resonant. In accelerator physics, an understanding of resonances and nonlinear dynamics is crucial for avoiding the loss of beam particles. Here we confirm the theoretical prediction of the dynamics for a single two-dimensional coupled resonance by observing so-called fixed lines. Specifically, we use the CERN Super Proton Synchrotron to measure the position of a particle beam at discrete locations around the accelerator. These measurements allow us to construct the Poincaré surface of section, which captures the main features of the dynamics in a periodic system. In our setting, any resonant particle passing through the Poincaré surface of section lies on a curve embedded in a four-dimensional phase space, the fixed line. These findings are relevant for mitigating beam degradation and thus for achieving high-intensity and high-brightness beams, as required for both current and future accelerator projects.

具有线性恢复力和反复出现的非线性扰动的系统的运动在物理学中至关重要。当系统的固有振荡频率和非线性恢复力的频率满足一定的代数关系时，动力学就会发生共振。在加速器物理学中，了解共振和非线性动力学对于避免束粒子损失至关重要。在这里，我们通过观察所谓的固定线来确认单个二维耦合共振动力学的理论预测。具体来说，我们使用 CERN 超级质子同步加速器来测量加速器周围离散位置的粒子束位置。这些测量使我们能够构建庞加莱截面曲面，它捕获了周期系统中动力学的主要特征。在我们的设置中，任何穿过庞加莱截面表面的共振粒子都位于嵌入四维相空间的曲线上，即固定线。这些发现与减轻光束退化有关，从而实现当前和未来加速器项目所需的高强度和高亮度光束。