Editing Foucault pendulum (section)

==Related physical systems==
[[File:Wheatstone Foucault device 256x256.png|thumb|The device described by Wheatstone.]]
Many physical systems precess in a similar manner to a Foucault pendulum. As early as 1836, the Scottish mathematician [[Edward Sang]] contrived and explained the precession of a spinning top.<ref>{{cite web | url=https://books.google.com/books?id=rtpQAAAAYAAJ&pg=PA105 | title=The Practical Mechanic's Journal | date=1857 }}</ref> In 1851, [[Charles Wheatstone]]<ref>[[Charles Wheatstone]] Wikisource: "[[s:Note relating to M. Foucault's new mechanical proof of the Rotation of the Earth|Note relating to M. Foucault's new mechanical proof of the Rotation of the Earth]]", pp. 65–68.</ref> described an apparatus that consists of a vibrating spring that is mounted on top of a disk so that it makes a fixed angle {{mvar|φ}} with the disk. The spring is struck so that it oscillates in a plane. When the disk is turned, the plane of oscillation changes just like the one of a Foucault pendulum at latitude {{mvar|φ}}.

Similarly, consider a nonspinning, perfectly balanced bicycle wheel mounted on a disk so that its axis of rotation makes an angle {{mvar|φ}} with the disk. When the disk undergoes a full clockwise revolution, the bicycle wheel will not return to its original position, but will have undergone a net rotation of {{math|2π sin ''φ''}}.

Foucault-like precession is observed in a virtual system wherein a massless particle is constrained to remain on a rotating plane that is inclined with respect to the axis of rotation.<ref name="psb1">{{cite arXiv |last=Bharadhwaj |first=Praveen |eprint=1408.3047 |class=physics.pop-ph |title=Foucault precession manifested in a simple system |year=2014 }}</ref>

Spin of a relativistic particle moving in a circular orbit precesses similar to the swing plane of Foucault pendulum. The relativistic velocity space in [[Minkowski spacetime]] can be treated as a sphere ''S''<sup>3</sup> in 4-dimensional [[Euclidean space]] with imaginary radius and imaginary timelike coordinate. Parallel transport of polarization vectors along such sphere gives rise to [[Thomas precession]], which is analogous to the rotation of the swing plane of Foucault pendulum due to parallel transport along a sphere ''S''<sup>2</sup> in 3-dimensional Euclidean space.<ref>{{cite journal | last1=Krivoruchenko | first1=M. I. | year=2009 | title=Rotation of the swing plane of Foucault's pendulum and Thomas spin precession: Two faces of one coin | journal=Phys. Usp. | volume=52 | issue=8| pages=821–829 |arxiv=0805.1136| bibcode=2009PhyU...52..821K| doi=10.3367/UFNe.0179.200908e.0873 | s2cid=118449576 }}</ref>

In physics, the evolution of such systems is determined by [[geometric phase]]s.<ref>"Geometric Phases in Physics", eds. [[Frank Wilczek]] and Alfred Shapere (World Scientific, Singapore, 1989).</ref><ref>L. Mangiarotti, G. [[Sardanashvily]], [https://books.google.com/books?id=-N6F44hlnhgC&dq=%22Berry+connection%22&pg=PA281 ''Gauge Mechanics''] (World Scientific, Singapore, 1998)</ref> Mathematically they are understood through parallel transport.