The presentation is divided in two parts :
(i) We model the momentum exchange in nonlinear wave-particle interaction using an N-body self-consistent Hamiltonian description. To drastically reduce the number of degrees of freedom, we use a discrete model allowing us to accurately study periodic structures, such as free electron lasers, gyrotrons or particle accelerators. From our model, we constructed an one-dimension time domain symplectic algorithm to simulate to metallic waveguides of traveling-wave tubes (TWTs). This algorithm is able to simulate arbitrary waveform (not just field envelope), including continuous waveform (CW), multiple carriers or digital modulations (telecom signals) and was validated against measurements.

(ii) Whenever light is slowed down, for any cause, two different formulas give its momentum (one kinematic and one canonical). For dielectrics, the coexistence of those momenta was the heart of the century-old Abraham-Minkowski dilemma, recently resolved. We demonstrate that this framework extends to momentum exchange in wave-particle interaction ; in particular to vacuum waveguides of electron tubes (metallic slow-wave structures). We also raise evidences that the dilemma occurs within the Landau damping (or amplification) context in plasmas.

Refs :
 D. F. G. Minenna, et al., Europhys. Lett., 122, 44002 (2018).
 D. F. G. Minenna, et al., arXiv : 1902.06431