Crab crossings are designed to increase the luminosity of accelerators by ensuring beam interactions are closer to a head on collision. One will be implemented at the Electron Ion Collider (EIC) at Brookhaven National Laboratory. It is then important to examine how the crab cavity will affect beam dynamics at the EIC. Methods such as Frequency Map Analysis (FMA) have been shown to be helpful in examining the phase space of accelerators in order to find properties such as resonances and the dynamic aperture. An alternative to such methods is an iterative method based on square matrix method that has been shown to reveal similar properties as FMA while reducing the computational power needed*,**. This method has been applied to the crab crossing scheme in order to find and explain effects of the higher order mode of crab cavities on the particle dynamics of the EIC.

Data driven methods have proved to be a useful tool for analyzing Hamiltonian systems. The symplectic condition is a strong constraint on Hamiltonian systems and it is therefore useful to implement this constraint into neural networks to ensure the accuracy of long term predictions about the system. One such method is the use of SympNets*, linear, activation, and gradient layers that guarantee the symplectic condition is met without the use of symplectic integration or extra gradient calculations. Data driven methods are also useful for calculating Koopman operators which aim to simplify nonlinear dynamical systems into linear ones. By using SympNets, one can ensure that the transformation described by the Koopman operator is symplectic, reversible, and more easily trained.

Crab crossings are designed to increase the luminosity of accelerators by ensuring beam interactions are closer to a head on collision. One will be implemented at the Electron Ion Collider (EIC) at Brookhaven National Laboratory. It is then important to examine how the crab cavity will affect beam dynamics at the EIC. Methods such as Frequency Map Analysis (FMA) have been shown to be helpful in examining the phase space of accelerators in order to find properties such as resonances and the dynamic aperture. An alternative to such methods is an iterative method based on square matrix method that has been shown to reveal similar properties as FMA while reducing the computational power needed*,**. This method has been applied to the crab crossing scheme in order to find and explain effects of the higher order mode of crab cavities on the particle dynamics of the EIC.