Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2402.16474

Particle Motion in Hamiltonian Formalism

Yannis Papaphilippou
View original source ↗Read source document ↗Export metadata ↓

Abstract

The goal of this contribution is to introduce the Hamiltonian formalism of theoretical mechanics for analysing motion in generic linear and non-linear dynamical systems, including particle accelerators. This framework allows the derivation and integration of equations of motion, in order to describe the particle trajectory evolution with respect to time. First, basic concepts are re-visited for describing particle propagation through the resolution of differential equations, applied to linear and non-linear motion. These equations of motion can be obtained by the Lagrangian of the system, which is the natural step leading to Hamiltonian Formalism and its properties. The accelerator ring Hamiltonian is derived, starting with the relativistic Hamiltonian of particles in the influence of E/M fields and a series of canonical (or symplectic) transformations and approximations. Thereby, introductory concepts of beam dynamics such as invariants and transport matrices are revisited and extended towards generic concepts such as action-angle variables and symplectic maps. To this end, the ground is prepared for the advanced methods and tools used for studying non-linear motion in particle accelerators.

Explore related subjects

physics.acc-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yannis Papaphilippou. 2024-02-26. Particle Motion in Hamiltonian Formalism. https://arxiv.org/abs/2402.16474

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Approximate solution of an adapted Bethe transport equation for electron scattering in foils

A new semi-analytical model for electron scattering in foils is presented valid for thin foils and electron-beam kinetic energies up to roughly $100$ keV and from 20 keV at a dimensionless foil thickness of $20$, and higher energies for thinner foils. To perform calculations, an approximate solution is constructed by Hankel transforms of an adapted Bethe transport equation for electron scattering. The Wentzel differential cross section is used in the Bethe transport Eq. and resulting integrals are solved analytically or approximated. These approximate solutions of the transport Eq.'s were compared to the Goudsmit-Saunderson solutions in gold, for thin samples with dimensionless thicknesses $λ=5,10,20$. We also compare to the Kawrakow \cite{Kawr} model, which uses the small angle approximation. We show that in the regime of $20$ to $100$ keV beam energies, our model performs better than Kawrakow's model and this is opposite above about 100 keV. Exactly, some angularly distributed beams and the influence of energy loss of the beam can be simulated with our model.

physics.acc-ph↗
arXiv2026

Reconstruction of Beam Transverse Parameters in the Fermilab Side-Coupled Linac Using a Normalized Coordinate Framework

Quadrupole scans are a commonly used tool for beam second moment reconstruction. Limitations in the strength of the magnets and layout of the beamline elements frequently preclude simple quadrupole-drift-detector scans from collecting sufficient data for reconstruction. Using a normalized coordinate framework, we characterize the prerequisites for a robust simple quadrupole scan and expand these prerequisites to reconstruction from more complex optics. The beam second moments are investigated at two locations in the Fermilab Side-Coupled Linac under simple and complex optics, using this framework to maximize information gained from wire scanner profile measurements.

physics.acc-ph↗
arXiv2026

Impedance of multilayer cylindrical structures with a material-filled beam region

Beam-coupling impedances in material media are relevant for ionization-cooling channels of a future muon collider, where the beam propagates in matter rather than vacuum. We extend the cylindrical field-matching formalism for multilayer structures to the case of a material-filled beam region surrounded by external layers with arbitrary electromagnetic properties. Relative to the vacuum formulation, the usual factor $1/γ^2$ is replaced by the material-dependent factor $F=1/\varepsilon_1-μ_1β^2$, and the radial propagation constant is modified accordingly. Analytical expressions are obtained for the monopolar longitudinal and dipolar transverse impedances, with the surrounding structure encoded through reflection coefficients determined by field matching. The formalism reduces to the known vacuum and perfectly conducting limits in the appropriate cases. Representative calculations are presented for absorber-relevant configurations, illustrating the dependence on the material properties of both the beam region and the surrounding layers. In particular, a finite beam-region conductivity can generate a real impedance component and reverse the sign of the imaginary space-charge impedance, while sufficiently high permittivity can give rise to resonant structures associated with oscillatory radial fields. The effect of finite-conductivity surroundings is also examined, showing the approach towards the perfectly conducting limit as the external conductivity is increased. The applicability of the infinite-length approximation is also discussed by comparison with finite-length mode-matching results.

physics.acc-ph↗