Appendix A - A digression on beam optics: stability, discrete formalism and chaotic behaviour

<!--HTML--><p>In&nbsp;this first Appendix we make a digression on the equations of motion that rule the oscillations of particles in a magnetic lattice. We first present the analogy with optics, showing the similarities and the differences with beam dynamics.</p> <p>We then introduce the discrete methods to solve the equations for particle oscillations. This formalism, based on maps or propagators (transfer matrix for the linear case), is different from the usual continuous approach of differential equations. Using this formalism we will show the stability of a sequence of focusing and defocusing quadrupoles.&nbsp;</p> <p>We conclude by addressing the problem of beam stability&nbsp;in an historical perspective. This question has the same mathematical formulation of the&nbsp;&nbsp;stability of the solar system, that was "the issue" in astronomy at the end of 19th century. We will recall the winding path that from Poincare' through the Russian school and to Lorenz has led to the discovery of chaotic behavior, and its implications for beam stability in a collider.</p>