Phase Portraits from Potential Energy - Bifurcations - Constraint Forces from Lagrange Multipliers

We first discuss the graphical method for sketching the 2-dimensional phase portrait, the representative trajectories, for a system which can be written as a second-order ODE coming from the negative gradient of a potential energy function (or effective potential energy function) for 1 degree of freedom. The equilibria are identified from 'hilltops' or 'valleys' of the potential energy curve and using the conserved effective energy, the remaining important features of the phase plane can be sketched..
When damping is included the phase portrait alters as stable equilibria become sinks, and have basins of attraction. We even show an experimental example..
We then introduce the idea of a bifurcation, which is qualitative change in system behavior as some parameter is varied. This can be understood in terms of qualitative changes in the phase portrait or changes in the potential energy. An example of a 'qualitative' change is, for example, when a stable equilibrium point becomes unstable, or if new equilibria emerge. .
We return to Lagrangian mechanics and introduce the form of constraint forces in the Lagrangian setting using Lagrange multipliers. A numerical example is considered. .
We provide a geometric interpretation of nonholonomic constraints in terms of local constraint surfaces. This allows us to .
understand parallel parking as a 'nonlinear' phenomenon and more generally, the problem of accessibility of configuration space when there are constraints..