多尺度法与准等时同步加速器粒子纵向运动的稳定性
Multi-scalar Techniques and Stabilities of Longitudinal Motion of Particles in Quasi-isochronous Synchrotron
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摘要: 在经典力学框架内和小振幅近似下， 把准等时同步加速器中的粒子纵向运动方程化为具有阻尼项、 受迫项的广义维尔斯特拉斯方程。在无扰动情况下， 用维尔斯特拉斯函数分析了系统的相平面特征； 在扰动情况下， 用多尺度法讨论了系统的稳定性。结果表明， 在相平面上， 分支轨道是一条过不稳定点的同宿轨道， 包围的区域呈“鱼形”或α形。系统的稳定性由“鱼形”区的面积决定， 面积越大系统越稳定； 结果还表明， 系统除了ωm=1的主共振外， 还存在ωm=2, 1/2的超次谐共振， 并找到了系统稳定性的临界条件。In the classical mechanics frame and with small amplitude approximation, the longutudial motion equation of particles in quasi\|isochronous syhchrotron is reduced to the general Weierstrass equation with a damping term and a forced term. In the non-perturbed case, the phase plane properties are analysed by using Weierstrass function; in the perturbed case, the stabilities are discussed in terms of the multi\|scalar techniques.The results show that the separatrix orbit is a homoclinic orbit through the instable point in the phase plane, the surrounding area is the fish form or α\|form.The stabilities are determined by the fish area, the large the area, the better the stability; also the results show that there are ωm=2, 1/2 super\| and sub\|harmonics resonance except the main resonance ωm=1, the critical condition of an instability is found.Abstract: In the classical mechanics frame and with small amplitude approximation, the longutudial motion equation of particles in quasi\|isochronous syhchrotron is reduced to the general Weierstrass equation with a damping term and a forced term. In the non-perturbed case, the phase plane properties are analysed by using Weierstrass function; in the perturbed case, the stabilities are discussed in terms of the multi\|scalar techniques.The results show that the separatrix orbit is a homoclinic orbit through the instable point in the phase plane, the surrounding area is the fish form or α\|form.The stabilities are determined by the fish area, the large the area, the better the stability; also the results show that there are ωm=2, 1/2 super\| and sub\|harmonics resonance except the main resonance ωm=1, the critical condition of an instability is found.
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