Instability of equilibrium points of some Lagrangian systems |
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Authors: | RS Freire Jr MVP Garcia FA Tal |
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Institution: | a Applied Math. Department, IME-USP, Brazil b Al. Joaquim Eugênio de Lima, 1055, apto. 31, 01403-003 São Paulo, Brazil |
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Abstract: | In this work we show that, if L is a natural Lagrangian system such that the k-jet of the potential energy ensures it does not have a minimum at the equilibrium and such that its Hessian has rank at least n−2, then there is an asymptotic trajectory to the associated equilibrium point and so the equilibrium is unstable. This applies, in particular, to analytic potentials with a saddle point and a Hessian with at most 2 null eigenvalues.The result is proven for Lagrangians in a specific form, and we show that the class of Lagrangians we are interested can be taken into this specific form by a subtle change of spatial coordinates. We also consider the extension of this results to systems subjected to gyroscopic forces. |
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Keywords: | Liapunov stability Lagrange-Dirichlet theorem Lagrangian systems |
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