Classical dynamics with curl forces, and motion driven by time-dependent flux

For position-dependent forces whose curl is non-zero ('curl forces'), there is no associated scalar potential and therefore no obvious Hamiltonian or Lagrangean and, except in special cases, no obvious conserved quantities. Nevertheless, the motion is nondissipative (measure-preserving in...

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Published inJournal of physics. A, Mathematical and theoretical Vol. 45; no. 30; pp. 305201 - 1-305201-18
Main Authors Berry, M V, Shukla, Pragya
Format Journal Article
LanguageEnglish
Published Bristol IOP Publishing 03.08.2012
IOP
Subjects
Betatrons
circulatory force
Exact sciences and technology
Flux
follower force
Magnetic flux
Mathematical analysis
Newtonian
non-Hamiltonian
nondissipative
Orbits
Origins
Physics
Position measurement
Rotational
Scalars
Magnetic flux
Aharonov-Bohm effect
Scalar potential
Dynamics
Hamiltonians
Positions
Orbits
Electric fields
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Summary:For position-dependent forces whose curl is non-zero ('curl forces'), there is no associated scalar potential and therefore no obvious Hamiltonian or Lagrangean and, except in special cases, no obvious conserved quantities. Nevertheless, the motion is nondissipative (measure-preserving in position and velocity). In a class of planar motions, some of which are exactly solvable, the curl force is directed azimuthally with a magnitude varying with radius, and the orbits are usually spirals. If the curl is concentrated at the origin (for example, the curl force could be an electric field generated by a changing localized magnetic flux, as in the betatron), a Hamiltonian does exist but violates the rotational symmetry of the force. In this case, reminiscent of the Aharonov-Bohm effect, the spiralling is extraordinarily slow.
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ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8113/45/30/305201