Nonlinear, completely integrable Hamiltonian systems that serve as blueprints for novel particle accelerators at the intensity frontier are promising avenues for research, as Fermilab’s Integrable Optics Test Accelerator (IOTA) example clearly illustrates. Here, we show that only very limited generalizations are possible when no approximations in the underlying Hamiltonian or Maxwell equations are allowed, as was the case for IOTA. Specifically, no such systems exist with invariants quadratic in the momenta, precluding straightforward generalization of the Courant–Snyder theory of linear integrable systems in beam physics. We also conjecture that no such systems exist with invariants of higher degree in the momenta. This leaves solenoidal magnetic fields, including their nonlinear fringe fields, as the only completely integrable static magnetic fields, albeit with invariants that are linear in the momenta. The difficulties come from enforcing Maxwell equations; without constraints, we show that there are many solutions. In particular, we discover a previously unknown large family of integrable Hamiltonians.

中文翻译：

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动量二次不变量的可积非线性磁场不存在


非线性、完全可积的哈密顿系统可以作为强度前沿新型粒子加速器的蓝图，是有前途的研究途径，正如费米实验室的可积光学测试加速器 (IOTA) 的例子清楚地说明了这一点。在这里，我们表明，当基础哈密顿量或麦克斯韦方程中不允许近似时，只能进行非常有限的概括，就像 IOTA 的情况一样。具体来说，不存在动量二次不变的系统，这妨碍了梁物理学中线性可积系统的库朗-斯奈德理论的直接推广。我们还推测不存在这样的系统，其动量具有更高程度的不变量。这使得螺线管磁场，包括它们的非线性边缘场，成为唯一完全可积的静磁场，尽管其动量是线性的不变量。困难来自于执行麦克斯韦方程组；在没有约束的情况下，我们表明有很多解决方案。特别是，我们发现了一个以前未知的可积哈密顿量大家族。