Mechanics analogue of the whirling geometric phase
In classical mechanics, the Hannay angle is a mechanics analogue of the whirling geometric phase (or Berry phase). It was named after John Hannay of the University of Bristol, UK. Hannay first described the angle in 1985, extending the ideas of the recently formalized Berry phase to classical mechanics.[1]
Consider a one-dimensional system moving in a cycle, like a pendulum. Now slowly vary a slow parameter , like pulling and pushing on the string of a pendulum. We can picture the motion of the system as having a fast oscillation and a slow oscillation. The fast oscillation is the motion of the pendulum, and the slow oscillation is the motion of our pulling on its string. If we picture the system in phase space, its motion sweeps out a torus.
The adiabatic theorem in classical mechanics states that the action variable, which corresponds to the phase space area enclosed by the system's orbit, remains approximately constant. Thus, after one slow oscillation period, the fast oscillation is back to the same cycle, but its phase on the cycle has changed during the time. The phase change has two leading orders.
The first order is the "dynamical angle", which is simply . This angle depends on the precise details of the motion, and it is of order .
The second order is Hannay's angle, which surprisingly is independent of the precise details of . It depends on the trajectory of , but not how fast or slow it traverses the trajectory. It is of order .[2]
^Hannay, J H (1985-02-01). "Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian". Journal of Physics A: Mathematical and General. 18 (2): 221–230. doi:10.1088/0305-4470/18/2/011. ISSN 0305-4470.
^Robbins, J M (2016-10-28). "The Hannay angle, thirty years on". Journal of Physics A: Mathematical and Theoretical. 49 (43): 431002. doi:10.1088/1751-8113/49/43/431002. hdl:1983/2992186e-5dde-4a3f-a2a9-67377afcadf9. ISSN 1751-8113.
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James p. 22 Hannah (2021), p. 120–124. Lee (2005), p. 278. Hannay (1911), p. 154. Hannay (1911), p. 153. Goodwin 2002, p. 259. Fremont-Barnes (2007)...
Interactions in Semiconductors" Chapter 5 pp. 192-221 in "Semiconductors" N. B. Hannay Ed. Reinhold, New York 1959 Light, Truman S.; Licht, Stuart; Bevilacqua...
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sentences incorporates text from a publication now in the public domain: Hannay, David McDowall (1911). "Navy". In Chisholm, Hugh (ed.). Encyclopædia Britannica...
(6172): 1191–1194. doi:10.1136/bmj.1.6172.1191. PMC 1599293. PMID 376051. Hannay, Richard (15 January 2013). "Royal Waterloo Hospital for Children and Women...
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UFRGS. Archived from the original on 2022-03-21. Retrieved 2021-10-21. Hannay, Peter (2017-01-01). "A non-device specific framework for the development...
London: Collins. pp. 247–248. ISBN 0-00-216548-1. Tute 1983, pp. 81–83 Hannay, David (1911). "Byng, John" . In Chisholm, Hugh (ed.). Encyclopædia Britannica...
Lawrence, chairman of Lloyd's Julian Mitchell, playwright David Hannay, Baron Hannay of Chiswick, ambassador to the United Nations Giles Radice, Labour...
– via Public Works and Government Services Canada, Translation Bureau. Hannay, Chris (1 January 2016). "Tory MP's bill seeks to clarify how dates are...
ISBN 1-84119-183-3 Media related to Battle of the Nile at Wikimedia Commons Hannay, David (1911). "Nile, Battle of the" . In Chisholm, Hugh (ed.). Encyclopædia...
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