We consider a Lagrangian Differential System (L.D.S.) with Lagrangian function L(q, q˙ )=T(q, q˙ )+U(q), sufficiently smooth in a neighbourhood of the critical pointq=0 of the potential functionU(q). The kinetic function T(q, q˙ ) is a non homogeneous quadratic function of the 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, q˙ )=(0,0), we provide a sufficient criterium for its instability.
Celletti, A., Negrini, P. (1994). On the instability of the equilibrium for a Lagrangian system with gyroscopic forces. NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 1, 313-322 [10.1007/BF01194983].
On the instability of the equilibrium for a Lagrangian system with gyroscopic forces
CELLETTI, ALESSANDRA;
1994-01-01
Abstract
We consider a Lagrangian Differential System (L.D.S.) with Lagrangian function L(q, q˙ )=T(q, q˙ )+U(q), sufficiently smooth in a neighbourhood of the critical pointq=0 of the potential functionU(q). The kinetic function T(q, q˙ ) is a non homogeneous quadratic function of the 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, q˙ )=(0,0), we provide a sufficient criterium for its instability.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
