On equilibrium instability for conservative and partially dissipative systems

Laloy, M.

doi:10.1016/0020-7462(76)90015-9

Cited by 11 publications

(3 citation statements)

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“…For instance, in [6], the author considered the kinetic energy T (q 1 , q 2 ,q 1 ,q 2 ) =q 2 (cos 1 q 2 + q 2 2 ). The associated system is such that the origin is an equilibrium point, π(0) = 0, π(q, q) < 0 if q = 0, and yet the equilibrium is stable.…”

Section: The Problemmentioning

confidence: 99%

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Instability of equilibrium points of some Lagrangian systems

Freire

Garcia

Tal

2008

Journal of Differential Equations

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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. The problemConsider the study of the Liapunov instability of equilibrium points of conservative Lagrangian systems in R 2n , with Lagrangians L(q,q) = T (q,q) − π(q), where π is the potential energy and T the kinetic energy.

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Section: The Problemmentioning

confidence: 99%

“…On the other hand, V < δq β n in C σ 3 ,λ and from the definition of V and the order of R we have Again, by(6), and recalling that from the (α, β)-, and it follows that (1 + ε)Finally, we can take σ 0 ∈ (0, σ 3 ] such that in C σ 0 ,λ…”

mentioning

confidence: 92%