Abstract
We consider a Lagrangian Differential System (L.D.S.) with Lagrangian function L(q,\(\dot q\))=T(q,\(\dot q\))+U(q), sufficiently smooth in a neighbourhood of the critical pointq=0 of the potential functionU(q). The kinetic function T(q,\(\dot q\)) is a non homogeneous quadratic function of the\(\dot q\)'s, i.e. the L.D.S. contains the so-called gyroscopic forces. The potential functionU(q) starts with a degenerate (but non zero), semidefinite-negative, quadratic form. Moreover,q=0 is not a proper maximum ofU, and this property has to be recognized in a suitable way. By analizing the problem of the existence of solutions of the L.D.S., which asymptotically tend to the equilibrium solution, (q,\(\dot q\))=(0,0), we provide a sufficient criterium for its instability.
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Work performed under the auspices of M.U.R.S.T. (Ministero dell'Universitá e della Ricerca Scientifica e Tecnologica) and G.N.F.M. (Gruppo Nazionale di Fisica Matematica of the National Research Council (C.N.R.)).
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Celletti, A., Negrini, P. On the instability of the equilibrium for a Lagrangian system with gyroscopic forces. NoDEA 1, 313–322 (1994). https://doi.org/10.1007/BF01194983
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DOI: https://doi.org/10.1007/BF01194983