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Gyroscopic stabilization of degenerate equilibria and the topology of real algebraic varieties. (English. Russian original) Zbl 1152.37022
Dokl. Math. 77, No. 3, 412-415 (2008); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 420, No. 4, 447-450 (2008).
Let \(x=(x_1,x_2,\ldots, x_n)\) be local coordinates of the state of \(n\) degrees of freedom mechanical system, \(T\) be kinetic energy – positive definite quadratic form with smooth coefficients \(T=\frac{1}{2}\sum g_{ij}(x)\dot{x}_i\dot{x}_j, V\) be potential energy – a smooth function of the system state, \(\omega\) be a smooth \(1\)-form specifying gyroscopic forces \(\omega=\sum \omega_i(x)\dot{x}_i.\) The equations of motion are
\[ [T]=-\frac{\partial V}{\partial x}-[\omega],\quad \text{where}\quad [f]=\left(\frac{\partial f}{\partial {\dot{x}}} \right)^{\cdot}-\frac{\partial f}{\partial {x}} \tag{1} \]
Let \(x=0\) be an equilibrium, i.e. (1) has the trivial solution \(x(t)=0,\dot{x}(t)=0.\) The equilibria coincide with critical points of the potential energy, so that \(dV(0)=0.\) Assume that \(V(0)=0.\) Let \(V=V_2+\ldots +V_m+\ldots\) be the Maclaurin series of \(V\), where \(V_m\) are homogeneous forms in \(x_1,\ldots,x_n\) of degree \(m\).
Equation (1) has a first integral – the energy integral \(T+V\). Thus the gyroscopic forces do not influence the conservation of the total energy, but they nay have a considerable effect on the stability of the equilibria. Particularly, the unstable equilibria of the system in a potential force field (when \(\omega=0\)) may become stable if suitable gyroscopic forces are added. This phenomenon is known as gyroscopic stabilization. If the quadratic form \(V_2\) is nongenerate then by the Morse lemma the potential energy in a neighborhood of \(x=0\) is reduced to \(\frac{1}{2}(x_1^2+\cdots +x_r^2-x_{r-1}^2-\cdots -x_n^2)\) and \(i=n-r\) is the index of inertia of \(V\) at the critical point \(x=0.\) The Thompson fundamental theorem asserts that an equilibrium can’t be stabilized by any gyroscopic force of the index of inertia is odd.
The aim of the article is to extend the Thompson theorem to degenerate equilibria, i.e. on the case of bifurcation points.

37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
70E05 Motion of the gyroscope
14P25 Topology of real algebraic varieties
14H70 Relationships between algebraic curves and integrable systems
Full Text: DOI
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