Earnshaw’s Theorem in Electrostatics and a Conditional Converse to Dirichlet’s Theorem

Jeffrey Rauch[1]

  • [1] Department of Mathematics University of Michigan Ann Arbor 48109 MI USA

Séminaire Laurent Schwartz — EDP et applications (2013-2014)

  • page 1-10
  • ISSN: 2266-0607

Abstract

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For the dynamics x ' ' = - ∇ x V ( x ) , an equilibrium point x ̲ are always unstable when on a neighborhood of x ̲ the non constant V satisfies P ( x , ∂ ) V = 0 for a real second order elliptic P . The proof uses a result of Kozlov [6].

How to cite

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Rauch, Jeffrey. "Earnshaw’s Theorem in Electrostatics and a Conditional Converse to Dirichlet’s Theorem." Séminaire Laurent Schwartz — EDP et applications (2013-2014): 1-10. <http://eudml.org/doc/275809>.

@article{Rauch2013-2014,
abstract = {For the dynamics $x^\{\prime \prime \} = -\nabla _xV(x)$, an equilibrium point $\underline\{x\}$ are always unstable when on a neighborhood of $\underline\{x\}$ the non constant $V$ satisfies $P(x,\partial )V=0$ for a real second order elliptic $P$. The proof uses a result of Kozlov [6].},
affiliation = {Department of Mathematics University of Michigan Ann Arbor 48109 MI USA},
author = {Rauch, Jeffrey},
journal = {Séminaire Laurent Schwartz — EDP et applications},
language = {eng},
pages = {1-10},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Earnshaw’s Theorem in Electrostatics and a Conditional Converse to Dirichlet’s Theorem},
url = {http://eudml.org/doc/275809},
year = {2013-2014},
}

TY - JOUR
AU - Rauch, Jeffrey
TI - Earnshaw’s Theorem in Electrostatics and a Conditional Converse to Dirichlet’s Theorem
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2013-2014
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 10
AB - For the dynamics $x^{\prime \prime } = -\nabla _xV(x)$, an equilibrium point $\underline{x}$ are always unstable when on a neighborhood of $\underline{x}$ the non constant $V$ satisfies $P(x,\partial )V=0$ for a real second order elliptic $P$. The proof uses a result of Kozlov [6].
LA - eng
UR - http://eudml.org/doc/275809
ER -

References

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  1. G. Allaire and J. Rauch, In preparation. 
  2. V. Arnold, Mathematical developments arising from Hilbert problems, Proceedings of Symposia in Pure Mathematics (F. Browder, Ed.), Amer. Math. Soc., Providence, R.I., 1976. MR419125
  3. S. Earnshaw, On the nature of the molecular forces which regulate the constitution of the luminferous ether, Trans. Cambridge Phil. Soc. 7, (1842) 97-112. 
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  5. V. Kozlov, Asymptotic solutions of equations of classical mechanics, J. Appl. Math. Mech. 46 (1982) 454-457. Zbl0522.70020MR714856
  6. V. Kozlov, Asymptotic motions and the inversion of the Lagrange-Dirichlet theorem, J. Appl. Math. Mech. 50 (1987) 719-725. Zbl0631.70018MR922185
  7. M. Laloy and K. Peiffer, On the instability of equilibrium when the potential has a non-strict local minimum, Arch. Rational Mech. Anal. 78 (1982) 213-222. Zbl0494.70021MR650844
  8. J. C. Maxwell, A Treatise on Electricity and Magnetism Vol. I., (From the 1891 ed.) Dover Publ. 1954. Zbl0056.20612MR63293
  9. P. Negrini, On the inversion of the Lagrange-Dirichlet Theorem, Resenhas 2 (1995), no. 1, 83-114. Zbl0848.34036MR1358331
  10. S. Taliaferro, Stability for two dimensional analytic potentials, J. Differential Equations 35 (1980) 248-265. Zbl0398.34047MR561980
  11. S. Taliaferro, Instability of an equilibrium in a potential field. Arch. Rational Mech. Anal. 109 no.2 (1990) 183-194. Zbl0679.70003MR1022514

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