Hi Sandor,
The differential length function of the loop is
D[ loop[u,n] ],u]
The length differential is
( (#.#)^(1/2) &) [ D[ loop[u,n] ],u] ] == ( ( 209 + 8 Cos[ n u] + Cos[2 n u])/2) ^(1/2)
This function is periodic with jumps by the winding length for a single loop.
Regards Roland
Sandor,
I am assuming you just want a numerical solution. Here is one simple way:
n=5;
lineLoop = ParametricPlot3D[{(2 + Cos[n u]) Cos[u], (2 + Cos[n u]) Sin[u], Sin[n u]}, {u, 0, 2 Pi}];
length = RegionMeasure@DiscretizeGraphics[lineLoop]
(* Out: 34.0784 *)
Regards – Henrik
ADDENDUM:
Well, I guess my approach above was a quite lazy one - things can be done better:
loopFunc[n_][t_] := {(2 + Cos[n t]) Cos[t], (2 + Cos[n t]) Sin[t], Sin[n t]};
ClearAll[v, n, t];
v[n_][t_] = D[loopFunc[n][t], t];
loopLength[n_?NumericQ] := NIntegrate[Norm[v[n][t]], {t, 0, 2 Pi}]
With this you get e.g. for winding number 5:
loopLength[5]
(* Out: 34.0869 *)
which is probably more exact than the other value above.
The loop length depending of the winding number is surprisingly not a monotone function:
(* length in terms of the respective full circle: *)
Plot[loopLength[n]/(6 Pi), {n, 0, 10}, PlotRange -> {0, Automatic}, GridLines -> {Range[10], None}]
FindMinimum[loopLength[n], n]
(* Out: {12.12186, {n -> 0.6715288}} *)
But one can convince oneself that this seems to be correct:
ParametricPlot3D[{loopFunc[0][t], loopFunc[1][t], loopFunc[2][t]}, {t,0, 2 Pi},
PlotLabels -> Placed[Style[#, 20] & /@ {"n = 0", "n = 1", "n = 2"}, Above]]
I dream of finding correlation between ratio of lengths and the ratio of elctron energy levels.
I dream of finding correlation between ratio of lengths and the ratio of elctron energy levels.
Hi Sandor, your dream has been resoved by Dirac and Hopf (and me).
The winding numbers of the Hopf bundle, the tangent bundle of unit Pauli matrices over the unit sphere, generate the 2n+1-Integer ladder of the spin eigenvalues of the angular momentum of a spinor field.
Dirac himself, Singer, Atiyha, Bott, Shapiro with quite another scope, generalized the Dirac equation on an compact metric n-manifolds systematically in their work to proof the index theorem, i.e. to find the number of independent, zero-divergence vector fields on a multiply connected manifold in the local data of the Laplacian.
The DIrac operator for electrons moving on a n-manifold is nothing else but the square root of the Laplacian plus a term proportional to the local curvature scalar.
Here is a paper dealing with the eigenvalues of the Dirac operator on a euclidean 2-torus.
Regards Roland
I dream of finding correlation between ratio of lengths and the ratio of elctron energy levels.
Sounds interesting - could you be more specific?