We propose a multi-particle self-consistent Hamiltonian (derived from an N-body description) that is applicable for periodic structures such as traveling-wave tubes (TWTs), gyrotrons, free-electron lasers, or particle accelerators. We build a 1D symplectic multi-particle algorithm to simulate the nonlinear wave-particle interaction in the time domain occurring in an experimental 3-meters long helix TWT. Our algorithm is efficient thanks to a drastic reduction model. A 3D helix version of our reduction model is provided. Finally, we establish an explicit expression of the electromagnetic power in the time domain and in non-monochromatic (non-continuous waveform) regime.
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