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Text 5. Lagrange—Dirichlet Criterion

For Hamiltonian systems in canonical form, an equilibrium point (qe, pe)is a point at which the partial derivatives of H vanish, that is, it is a critical point of H. If the 2n x 2n matrix δ2H of second partial derivatives evaluated at (qe, pe) is positive or negative definite (that is, all the eigenvalues of δ2H(qe, pe) have the same sign), then (qe, pe) is stable. This follows from conservation of energy and the fact from calculus that the level sets of H near (qe, pe)are approximately ellipsoids. As mentioned earlier, this condition implies, but is not implied by, spectral stability. The KAM (Kolmogorov, Arnold, Moser) theorem, which gives stability of periodic solutions for two-degree-of-freedom systems, and the Lagrange-Dirichlet theorem are the most basic general stability theorems for equilibria of Hamiltonian systems.

For example, let us apply the Lagrange-Dirichlet theorem to a classical mechanical system whose Hamiltonian has the form kinetic plus potential energy. If (qe, pe)is an equilibrium, it follows that pe is zero. Moreover, the matrix δ2H of second-order partial derivatives of H evaluated at (qe, Pe)block diagonalizes, with one of the blocks being the matrix of the quadratic form of the kinetic energy, which is always positive definite. Therefore, if δ2H is definite, it must be positive definite, and this in turn happens if and only if δ2V is positive definite at qe, where V is the potential energy of the system. We conclude that for a mechanical system whose Lagrangian is kinetic minus potential energy, (qe,0) is a stable equilibrium, provided that the matrix δ2V(qe) of second-order partial derivatives of the potential V at qe is positive definite (or, more generally, qe is a strict local minimum for V). If δ2V at qe has a negative definite direction, then qe is an unstable equilibrium.

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