# Hamiltonian Analysis of the Particle Motion in an Accelerator with the Longitudinal Magnetic Field

## Abstract

The particle motion at a presence of a large magnetic field directed along the particle trajectory demands the special description. This article deals with the decomposition of the Hamiltonian on the two parts: fast and slow motion. The first part describes the fast rotation around the magnetic line of longitudinal field. The second part describes the slow drift of rotation center from one magnetic line to another. The supposed method enables to write the simple Hamiltonian to each motion type and to formulate the matrix formalism for any element of an accelerator device (quadruple, skew- quadruple, drift gap, bend with a filed index). The Hamiltonian decomposition has physical clearness when the longitudinal field is larger than another fields but it is correct for the arbitrary parameters. At the small longitudinal field the coupling term in Hamiltonian between two modes is essential. The dispersion property of fast and slow modes is derived easy from Hamiltonian also. This method expands easily for nonlinear motion of such modes. This results may be used at analyzed the electron motion in the cooling device, the muon motion in the muon ionization cooler or another system with strong solenoidal coupling.

- Authors:

- Budker Institute of Nuclear Physics, Novosibirsk (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 20798392

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: AIP Conference Proceedings; Journal Volume: 821; Journal Issue: 1; Conference: COOL05: International workshop on beam cooling and related topics, Galena, IL (United States), 18-23 Sep 2005; Other Information: DOI: 10.1063/1.2190107; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS; ACCELERATORS; ADIABATIC DEMAGNETIZATION; BEAM DYNAMICS; COUPLING; ELECTRON BEAMS; ELECTRONS; HAMILTONIANS; MAGNETIC FIELDS; MUONS; NONLINEAR PROBLEMS; NUMERICAL ANALYSIS; ROTATION; TRAJECTORIES

### Citation Formats

```
Reva, V. B..
```*Hamiltonian Analysis of the Particle Motion in an Accelerator with the Longitudinal Magnetic Field*. United States: N. p., 2006.
Web. doi:10.1063/1.2190107.

```
Reva, V. B..
```*Hamiltonian Analysis of the Particle Motion in an Accelerator with the Longitudinal Magnetic Field*. United States. doi:10.1063/1.2190107.

```
Reva, V. B.. Mon .
"Hamiltonian Analysis of the Particle Motion in an Accelerator with the Longitudinal Magnetic Field". United States.
doi:10.1063/1.2190107.
```

```
@article{osti_20798392,
```

title = {Hamiltonian Analysis of the Particle Motion in an Accelerator with the Longitudinal Magnetic Field},

author = {Reva, V. B.},

abstractNote = {The particle motion at a presence of a large magnetic field directed along the particle trajectory demands the special description. This article deals with the decomposition of the Hamiltonian on the two parts: fast and slow motion. The first part describes the fast rotation around the magnetic line of longitudinal field. The second part describes the slow drift of rotation center from one magnetic line to another. The supposed method enables to write the simple Hamiltonian to each motion type and to formulate the matrix formalism for any element of an accelerator device (quadruple, skew- quadruple, drift gap, bend with a filed index). The Hamiltonian decomposition has physical clearness when the longitudinal field is larger than another fields but it is correct for the arbitrary parameters. At the small longitudinal field the coupling term in Hamiltonian between two modes is essential. The dispersion property of fast and slow modes is derived easy from Hamiltonian also. This method expands easily for nonlinear motion of such modes. This results may be used at analyzed the electron motion in the cooling device, the muon motion in the muon ionization cooler or another system with strong solenoidal coupling.},

doi = {10.1063/1.2190107},

journal = {AIP Conference Proceedings},

number = 1,

volume = 821,

place = {United States},

year = {Mon Mar 20 00:00:00 EST 2006},

month = {Mon Mar 20 00:00:00 EST 2006}

}