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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 …
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Classroom Contents
Phase Portraits from Potential Energy - Bifurcations - Constraint Forces from Lagrange Multipliers
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- 1 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 negativ…
- 2 When damping is included the phase portrait alters as stable equilibria become sinks, and have basins of attraction. We even show an experimental example.
- 3 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 …
- 4 We return to Lagrangian mechanics and introduce the form of constraint forces in the Lagrangian setting using Lagrange multipliers. A numerical example is considered.
- 5 We provide a geometric interpretation of nonholonomic constraints in terms of local constraint surfaces. This allows us to
- 6 understand parallel parking as a 'nonlinear' phenomenon and more generally, the problem of accessibility of configuration space when there are constraints.
