Concatenation of Two Different Integrable Systems is Not Integrable

Aims/Objectives: Nonlinear, completely integrable Hamiltonian systems representing charged particle motion in external electromagnetic fields hold promise for models of novel intensity frontier particle accelerators. The main reason is the combination of large regions of stable orbits with damping of collective instabilities by conservative relaxation. Intensity frontier particle accelerators are essential for discovery science in particle and nuclear physics, and a host of industrial and security applications.Study Design: Mathematical proof.Methodology: Tools of Hamiltonian dynamics and differential geometry.Results: Realistic system lattices include additional sections or inserts that themselves may be integrable (such as linear optics, phase trombones, thin lenses, kicks, etc.). However, in general the full system fails to remain integrable.Conclusion: The non-integrability proof is presented and some consequences of integrability failure are explored.