English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We return to our study \cite{BEH} of invariant spin fields and spin tunes forpolarized beams in storage rings but in contrast to the continuous-timetreatment in \cite{BEH}, we now employ a discrete-time formalism, beginningwith the $\rm{Poincar\acute{e}}$ maps of the continuous time formalism. We thensubstantially extend our toolset and generalize the notions of invariant spinfield and invariant frame field. We revisit some old theorems and prove severaltheorems believed to be new. In particular we study two transformation rules,one of them known and the other new, where the former turns out to be an$SO(3)$-gauge transformation rule. We then apply the theory to the dynamics ofspin-$1/2$ and spin-$1$ particle bunches and their density matrix functions,describing semiclassically the particle-spin content of bunches. Our approachthus unifies the spin-vector dynamics from the T-BMT equation with thespin-tensor dynamics and other dynamics. This unifying aspect of our approachrelates the examples elegantly and uncovers relations between the variousunderlying dynamical systems in a transparent way. As in \cite{BEH}, theparticle motion is integrable but we now allow for nonlinear particle motion oneach torus. Since this work is inspired by notions from the theory of bundles,we also provide insight into the underlying bundle-theoretic aspects of thewell-established concepts of invariant spin field, spin tune and invariantframe field. Since we neglect, as is usual, the Stern-Gerlach force, theunderlying principal bundle is of product formso that we can present the theoryin a fashion which does not use bundle theory. Nevertheless we occasionallymention the bundle-theoretic meaningof our concepts and we also mention thesimilarities with the geometrical approach to Yang-Mills Theory.

A detailed and unified treatment of spin-orbit systems using tools distilled from the theory of bundles