!!§!! ICCTPEA-2014, St.Petersburg
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On the Stationary Points
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of Coulomb’s Potential Generated by Point Charges
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Alexei Uteshev
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St.Peterburg State University
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~~TOC~~
==Problem==
Point charges: equilibrium positions for
$$
F(P)=\frac{m_1}{|PP_1|}+\dots+ \frac{m_K}{|PP_K|}
$$
Coulomb's
or
Newton's
potentials
$$ \{P_1,\dots,P_K\} \subset \mathbb R^n,\ \{m_1,\dots,m_K\} \subset \mathbb R $$
$$ | \cdot | - \mbox{ Euclidean metrics } $$
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Maxwell's
conjecture:
The number of stationary points
$ \le (K-1)^2 $.
[Not proved].
==Difficulties==
Coulomb's potential
: $ n=2,\ K= 3 $;
stationary points
{{ algebra2:optimiz:distance:coulomb_50.jpg |}}
$$ F(P)=\frac{m_1}{|PP_1|}+\frac{m_2}{|PP_2|}+\frac{m_3}{|PP_3|} $$
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Gradient systems
$$
\frac{D\, F}{D\, P} = \mathbb O
$$
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Reduction to an algebraic system
$$
\begin{array}{c}
\left\{\begin{array}{l}
\displaystyle \frac{m_1(x-x_1)}{\sqrt{(x-x_1)^2+(y-y_1)^2}^3}+ \frac{m_2(x-x_2)}{\sqrt{(x-x_2)^2+(y-y_2)^2}^3}+ \frac{m_3(x-x_3)}{\sqrt{(x-x_3)^2+(y-y_3)^2}^3} = 0, \\
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\displaystyle \frac{m_1(y-y_1)}{\sqrt{(x-x_1)^2+(y-y_1)^2}^3}+ \frac{m_2(y-y_2)}{\sqrt{(x-x_2)^2+(y-y_2)^2}^3}+ \frac{m_3(y-y_3)}{\sqrt{(x-x_3)^2+(y-y_3)^2}^3} = 0
\end{array} \right. \\
\Downarrow \\
{\rm Squaring \ procedure} \\
\Downarrow \\
\left\{
\begin{array}{ll}
F_1(x,y,m_1,m_2, m_3) = 0, \\
F_2(x,y,m_1, m_2,m_3) = 0,
\end{array}
\right. \deg_{[x,y]} F_j = 28 \\
\Downarrow \\
{\rm Elimination \ of \ variable} \\
\Downarrow \\
...
\end{array}
$$
T
:-(:-( hard
!!!
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==Bypass==
Inverse problem
:
Given
$$ P_1,\dots,P_{n+1} , P_{\ast} \subset \mathbb R^{n} $$
find
$ \{m_1,\dots,m_{n+1}\} $
for
$ P_{\ast} $
to be a stationary point for
$$ F(P)=\sum_{j=1}^{n+1} m_j |PP_j|^{-1} $$
Essential:
Number of points $ = 1+ \dim $:
$$ K=n+1 \ . $$
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==n=2==
$$ \{ m_j^{\ast} = |P_{\ast}P_j|^{3} S_j \}_{j=1}^3 $$
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$$
S_1=\left|
\begin{array}{ccc}
1 & 1 & 1 \\
x_{\ast} & x_2 & x_3 \\
y_{\ast} & y_2 & y_3
\end{array}
\right|,\
S_2=
\left|
\begin{array}{ccc}
1 & 1 & 1 \\
x_1 & x_{\ast} & x_3 \\
y_1 & y_{\ast} & y_3
\end{array}
\right|,\ S_3 =
\left|
\begin{array}{ccc}
1 & 1 & 1 \\
x_1 & x_2 & x_{\ast} \\
y_1 & y_2 & y_{\ast}
\end{array}
\right|
$$
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!!Th!! $ P_{\ast}=(x_{\ast},y_{\ast}) $
is a stationary point for
$ F(P)=m_1^{\ast}|PP_1|^{-1} +m_2^{\ast}|PP_2|^{-1} +m_3^{\ast}|PP_3|^{-1} $.
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$$
\left\{
\begin{array}{ccc}
m_1|PP_1|^{-3} (x-x_1)+m_2|PP_2|^{-3} (x-x_2)+m_3|PP_3|^{-3} (x-x_3)&=&0, \\
m_1|PP_1|^{-3} (y-y_1)+m_2|PP_2|^{-3} (y-y_2)+m_3|PP_3|^{-3} (y-y_3)&=&0. \\
\end{array}
\right.
$$
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Linear system w.r.t.
$ m_1,m_2,m_3 $
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$$ m_1:m_2:m_3= |PP_1|^{3} S_1(x,y) : |PP_2|^{3} S_2(x,y) : |PP_3|^{3} S_3(x,y) $$
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$$
\left[\frac{m_2S_1}{m_1S_2} \right]^2= \left[\frac{|PP_2|}{|PP_1|}\right]^{6},\ \left[\frac{m_2S_3}{m_3S_2} \right]^2= \left[\frac{|PP_2|}{|PP_3|} \right]^{6} \ .
$$
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==Coulomb 3 charges: bifurcation picture==
!!Ex.!!
$$
F(x,y)=\frac{1}{\sqrt{(x-1)^2+(y-1)^2}}+ \frac{m_2}{\sqrt{(x-5)^2+(y-1)^2}}+\frac{m_3}{\sqrt{(x-2)^2+(y-6)^2}}
$$
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{{ algebra2:optimiz:distance:torri:3weights1.png |}}
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$$
\begin{array}{c}
\widetilde F_1(x,y,m_2,m_3)= \\
=m_2^2\,(5\,x+3\,y-28)^2(x^2+y^2-2\,x-2\,y+2)^3 -(5\,x-y-4)^2(x^2+y^2-10\,x-2\,y+26)^3 , \\
\widetilde F_2(x,y,m_2,m_3)= \\
=m_2^2\,(4\,y-4)^2(x^2+y^2-4\,x-12\,y+40)^3 -m_3^2\,(5\,x-y-4)^2(x^2+y^2-10\,x-2\,y+26)^3.
\end{array}
$$
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Find the locus of stationary point for any
$ m_2, m_3 $:
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$$ \widetilde F_1(x,y,m_2) = 0 $$
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{{ algebra2:optimiz:distance:torri:figure011.png |}}
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$$
\mathbf{Resultant}_x (\widetilde F_1, \widetilde F_2)\equiv (y-1)^8(y-6)^4 \mathcal Y(y,m_2,m_3), \ \deg_y \mathcal Y =34 \ .
$$
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2
vs.
4
stationary point
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$$
\mathbf{Discriminant}_y( \mathcal Y(y,m_2,m_3)) \equiv
\Xi^2(m_2,m_3) \Psi(m_2,m_3) \quad , \quad \deg \Xi=444, \deg \Psi =48
$$
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$$
\Psi(m_2,m_3)=
$$
$$
=3^{36}(169\,m_2^2+192\,m_2m_3+64\,m_3^2)^5(169\,m_2^2 -192\,m_2m_3+64\,m_3^2)^5(28561\,m_2^4+19968\,m_2^2m_3^2+4096\,m_3^4)^7 $$
$$ + \dots + $$
$$ + 2^2\cdot 3^{31} \cdot 17^{40} (5545037166327\, m_2^4-161882110764644\,m_2^2m_3^2+1656772227072\,m_3^4) $$
$$ + 2^3\cdot 3^{36}\cdot 17^{44} (51827\,m_2^2+28112\,m_3^2)+ 3^{36}\cdot 17^{48} \ . $$
{{ algebra2:optimiz:distance:torri:stability_param13.png |}}
Conditions for
$ \exists $
4
stationary points
<=>
Conditions for
$ \exists $
1
stable
stationary point
{{ algebra2:optimiz:distance:torri:green03.png |}}
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?
Where is the locus of stable equilibrium
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!!Th.!!
If
$ (m_1,m_2,m_2) \in $
stability domain in parameter space
=>
stable stationary point
$ \in $
$$ \Phi(x,y) > \frac{2}{9} S^2 \ . $$
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$$
\Phi(x,y)= \frac{S_1(x,y)S_2(x,y)S_3(x,y)}{|PP_1|^2 |PP_2|^2 |PP_3|^2}
\left|
\begin{array}{cccc}
1 & 1 & 1 & 1\\
x& x_1 & x_2 & x_3 \\
y& y_1 & y_2 & y_3 \\
x^2+y^2 & x_1^2+y_1^2 & x_2^2+y_2^2 & x_3^2+y_3^2
\end{array}
\right| \ .
$$
with
$$
S_1(x,y)=\left|
\begin{array}{ccc}
1 & 1 & 1 \\
x & x_2 & x_3 \\
y & y_2 & y_3
\end{array}
\right|, \
S_2(x,y)=\left|
\begin{array}{ccc}
1 & 1 & 1 \\
x_1 & x & x_3 \\
y_1 & y & y_3
\end{array}
\right|,\
S_3(x,y)=\left|
\begin{array}{ccc}
1 & 1 & 1 \\
x_1 & x_2 & x \\
y_1 & y_2 & y
\end{array}
\right|
$$
$$
S=S_1+S_2+S_3=
\left|
\begin{array}{ccc}
1 & 1 & 1 \\
x_1 & x_2 & x_3 \\
y_1 & y_2 & y_3
\end{array}
\right|
$$
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!!Ex.!!
$$ \begin{array}{c} P_1 =(1,1) \\ m_1 \end{array} , \begin{array}{c} P_2=(5,1) \\ m_2 \end{array} , \begin{array}{c} P_3= (2,6) \\ m_3 \end{array} $$
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$$
\Phi(x,y)=\frac{16(28-5\,x-3\,y)(5\,x-y-4)(y-1)(-52+30\,x+32\,y-5\,x^2-5\,y^2)}{((x-1)^2+(y-1)^2)((x-5)^2+(y-1)^2)((x-2)^2+(y-6)^2)} \ .
$$
{{ algebra2:optimiz:distance:torri:attraction_domain3.png |}}
{{ algebra2:optimiz:distance:torri:attraction_domain2.png |}}
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==3D case (n=3)==
!!Th.!!
Earnshaw (1842).
Impossible to stabilize equlibrium.
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Maxwell's
conjecture:
The number of stationary points for
$ K=4 $
charges
$ \le 9 $.
[Not proved].
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Progress
: $ \le 7 $
==References==
[1]. **Maxwell J.C.** //A Treatise on Electricity and Magnetism.// Vol. 1. Dower, New York. 1954
[2]. **Uteshev A.Yu.** //Analytical Solution for the Generalized Fermat-Torricelli Problem//. Amer.Math.Monthly. **121**, N 4, 318-331, 2014. Preprint ☞ ((http://arxiv.org/pdf/1208.3324v1.pdf arXive:1208.3324v1)) (220 Kb)
[3]. **Uteshev A.Yu., Yashina M.V.** //Stationary Points for the Family of Fermat-Torricelli-Coulomb-like potential functions//. Proc. 15th Workshop CASC (Computer Algebra in Scientific Computing), Berlin 2013. Springer. LNCS. **8136**, 412-426, 2013.