Instability of equilibrium points of some Lagrangian systems

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. 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.

A Partial Reciprocal of Dirichlet Lagrange Theorem Detected by Jets

“…, q n ), where 0 ≤ r < n, π s is an homogeneous polynomial of degree s ≥ 2 which does not have a minimum at the origin, and R obeys lim q→0 R(q) q s = 0. Other relevant works on this conjecture were made by Freire, Garcia and Tal in [2,7], in the first of these articles the conjecture is proved in the case of systems with two degrees of freedom, the second one studies the general case n degrees of freedom and it is shown that if the punctual jet of order s of at 0, j s , shows that have not minimum at origin and the Hessian of in 0 is a positive semi-definite quadratic form whose kernel has dimension at most 2, we have the instability of the origin.…”

Kinetic Energy and the Stable Set

“…Taliaferro [3] proved that all weak and strong maximum points are unstable if H ∈ C 1 and Palamodov [4] proved that non-minimum points are unstable under the hypothesis of analyticity. Many other relevant partial results were obtained, a good number assuming non-degeneracy conditions or restrictions to the degrees of freedom (see for instance [5][6][7][8]). A common point to these results is that the instability of the equilibrium is proved using different hypothesis on the potential energy but no non-standard hypothesis on the kinetic energy.…”