On a lagrangean action based kinetic instability theorem of Kelvin and Tait

This article provides a simple and comprehensive proof of the Kelvin/Tait Kinetic Instability theorem. According to it: “if the Lagrangean Action (LA) of a holonomic and conservative system along a fundamental trajectory $\text{I}$ for cotermini and noncontemporaneous but isoenergetic variations $\text{}\text{̂}\text{gd(I)}$, is a minimum, when calculated from an initial configuration to a final one reached at any time, however much later, then $\text{I}$ is unstable”. The proof hinges on the study of the sign of the second such variation $\text{}\text{̂}\text{gd}{}^2\text{(LA)}$, of $\text{LA}$. The introduction of the concept of Lagrangean kinetic foci, combined with the extended Jacobi's sufficiency condition (for this variable endpoints variational problem) leads to a hitherto missing quantitative formulation of the theorem. Possible extensions to nonconservative systems are also indicated.