Scope of the quadratic-mode and energy-stability results: exact quadratic model versus linearisation approximation

The three results of this group establish different, and differently qualified, statements.

1. Decoupling of the constant-coefficient quadratic system $M{q}^{\mathrm{\prime }\mathrm{\prime }}+Kq=0$ into normal modes, with the zero-mode and energy identities is an exact theorem about the constant-coefficient linear system $M{q}^{\mathrm{\prime }\mathrm{\prime }}+Kq=0$ with $M$ symmetric positive definite and $K$ symmetric non-negative: it diagonalises the system, gives all solutions and the modal energies. It is not an approximation statement, and it makes no claim about nonlinear systems or about configuration-dependent kinetic metrics.

2. Linearisation of the mechanical system at an equilibrium: M(q_0) xi'' + D²U(q_0) xi = 0, and the small-oscillation quadratic model computes the derivative of the nonlinear mechanical field at an equilibrium. Its linearised equation is the exact first-order Taylor model at that point; replacing the nonlinear system by it is an approximation whose error over a time interval is not estimated by the proposition. A zero eigenvalue ${\lambda }_i=0$ of $M(q_0{)}^{-1\mathrm{/}2}{D}^{2}U(q_0)M(q_0{)}^{-1\mathrm{/}2}$ corresponds to an affine mode of the linearisation, not to an oscillation, and a negative eigenvalue corresponds to exponential growth of the linearisation.

3. A strict local minimum of the potential gives a Lyapunov stable equilibrium of the mechanical system, by energy conservation is an exact Lyapunov-stability theorem for the nonlinear system at a strict minimum of the potential. It is a statement about the nonlinear field of The mechanical first-order field, Lagrangian and energy of a configuration-dependent kinetic metric in a chart, uses exact energy conservation, and therefore proves stability but not attraction: the theorem does not assert that solutions converge to the equilibrium, and in a conservative model with no dissipation they do not in general. It also does not assert that the linearisation determines the nonlinear stability in non-hyperbolic cases, and it does not require the Hessian to be positive definite.