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Knots & Classical Electrodynamics

Invited Lecture
by
Samuel J. Lomonaco, Jr. ^(*)
at
Short Course on Knots & Physics
Annual Meeting
American Mathematical Society
San Francisco, California
January, 1995

Abstract

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Samuel J. Lomonaco, Jr.
E-mail: Lomonaco@UMBC.EDU

Key words and Phrases. Lord Kelvin, Sir William Thomson, Peter Guthrie Tait, Knots, Electrodynamics, Electromagnetism, Energy, Magnetic Energy, Electrostic Energy, Magnetic Knots, Electrostatic Knots, Minimal Energy Knots, Vortices, Helicity, Magnetic Surfaces, Asymptotic Behavior Objective, Maxwell's Equations, Plasma Physics, Chern-Simon action, Gauge Theory

A Search for a Title

Preamble

Slide 1 The title is ...

Slide 2 On the other hand, the title could have been ...

Slide 3 But the title really should be ...

Slide 4 Acknowledgement

Back to the 19-th Century

Introduction

Slide 0 A letter from J. Clerk Maxwell to Peter Guthrie Tait

Slide 1 The state of electromagnetism before J. Clerk Maxwell

Slide 1a The state of electromagnetism after J. Clerk Maxwell

Slide 2 Conventions adopted during this talk

Slide 3 A chain of events beginning with Herman von Helmholtz

Slide 4 Tait's vortex machine

Slide 4a Tait's lecture with his vortex machine

Slide 5 While attending Tait's lecture, Sir William Thomson (Lord Kelvin) conceived the atomic vortex theory

Slide 6 This fits within the Maxwell Milieu

Slide 7 Another look at Maxwell's letter to Tait

Slide 8 Exactly what was Maxwell saying in his letter?

Slide 9 Tait attempted to correlate knot types with the chemical elements

Slide 9a Tait's Periodic Table

Slide 10 The demise of Thomson's (Lord Kelvin's) atomic vortex theory

Slide 11 Why does the atomic vortex theory still persist?

Slide 12 Why such longevity?

Back to the 20-th Century

Modern Day Magnetic Vortices

Slide 1 Modern day magnetic vortices

Slide 2 Ideal magnetohydrodynamics

Slide 3 Consider an ideal constant density, incompressible, perfectly conducting fluid with a magnetic field B

Slide 4 Equations which govern such a fluid

Slide 5 The magnetic vector potential A

Slide 6 Consequences of incompressibility

Slide 7 The invariance of the volume of a closed surface moving with the fluid

Slide 8 Frozen field effects

Slide 9 Velocity of the magnetic lines of force

Slide 10 Definition of a magnetic surface

Slide 11 Volume is an invariant of a magnetic surface as it moves with the fluid

Slide 12 The magnetic flux enclosed by a magnetics surface is an invariant

Slide 13 The standard torus

Slide 14 The standard foliation of the standard torus

Slide 15 The use of Dehn surgery to construct other foliations of the standard torus

Slide 16 Magnetic knots & links

Slide 17 Two Examples: A magnetic Hopf link and a magnetic trefoil

Slide 18 The boundary of a magnetic knot/link is a magnetic surface. Moreover, it remains a magnetic surface as it changes with the flow.

Slide 19 Some flow invariants: Knot/Link type, enclosed volume, and enclosed flux

Slide 20 Another flow invariant: The linking numbers LK_ij(g_tL)

Slide 21 Another flow invariant: The self-linking number

Slide 22 Summary of flow invariants so far

Slide 23 Another flow invariant: The helicity

Slide 23a The helicity is the same as the Chern-Simon action

A digression: Maxwell's equations expressed in terms of differential forms

Slide 23b Electromagnetic 1-, 2-, and 3-forms
Slide 23c Maxwell's equations written in terms of differential forms
Slide 23d Maxwell's equations in an even more concise form

Slide 24 Helicity as a function of Self-linking and linking numbers

Slide 25 An example: The helicity of a magnetic trefoil from above formula

Slide 26 Helicity of a magnetic Hopf link from above formula

Slide 27 The energy of a magnetic link

Slide 28 Consider a magnetic link in a constant density, incompressible, perfectly conducting, viscous fluid

Slide 29 A description of the energy dissipation of a magnetic knot in a viscous fluid

Slide 30 Energy dissipation terminates because of topology

Slide 31 An example of the energy dissipation of a Magnetic trefoil in a viscous fluid being bounded by topology

Slide 32 How do we quantify the concept of topology bounding energy dissipation?

Slide 32a An informal definition of the asymptotic crossing number

Slide 33 Keith Moffatt's energy spectrum of knots

Slide 34 The energy spectrum of a knot (Cont.)

Slide 35 How do we compute the knot invariants arising from their energy spectrum?

Back to the 20-th Century, Again

Modern Day Electrostatic Vortices

Slide 1 Knotted electrostatic vortices

Slide 2 Conventions assumed during the talk

Slide 3 A charged knotted wire assuming minimal energy position. THONG!

Slide 3a A possible example of a minimal energy electrostatic link

Slide 3b A possible example of a minimal energy electrostatic knot

Slide 4 The honey jar problem

Slide 5 Types of honey jar problems

Slide 6 Types of honey jar problems (Continued)

Slide 7 The Asymptotic Behavior (AB) Objective

Slide 8 Electrostatic energy

Slide 9 The honey jar problem for curves

Slide 10 But the electrostatic energy of a curve is always infinite!

Slide 11 Renormalization and some equations for minimal e-energy knots

Slide 12 Minimal e-energy equations( Continued)

Slide 13 Asymptotic behavior of the equations

Slide 14 The Asymptotic Behavior Objective (AB) yardstick?

Slide 15 Freedman-He-Wang renormalization

Slide 16 A theorem of Freedman & He

Slide 17 Mobius "energy", a non-physical energy. The AB objective yardstick is abandoned.

Slide 17a A reminder of our original motivation

Slide 18 Theorems of Freeman & He on knots/links of minimal Mobius energy

Slide 19 Knots/links of minimal Mobius energy (Continued)

Slide 20 The honey jar problem for hollow & solid tubes. No need for renormalization.

Slide 21 Minimal energy equations for the conducting honey jar problem for hollow tubes

Slide 22 Minimal energy equations (Continued)

Slide 23 Minimal energy equations for honey jar problem for hollow & solid tubes

And on to the Next Millennium ...

The End

Paper based on this talk: The modern legacies of Thomson's atomic vortex theory in classical electrodynamics, in "The Interface of Knots and Physics," edited by L.H. Kauffman, AMS PSAPM, Vol. 51, Providence, RI (1996), pp. 145 - 166.

In postscript form (10.6MB) (22pages)
In gzipped postscript form (1.2MB) (22 pages)
The paper in PDF form (2.0MB) (22 pages)
Web HomePage for Samuel J. Lomonaco, Jr.

(*) Partially supported by the L-O-O-P Fund.

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