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

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

Ross Dynamics Lab via YouTube Direct link

We return to Lagrangian mechanics and introduce the form of constraint forces in the Lagrangian setting using Lagrange multipliers. A numerical example is considered.

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4 of 6

We return to Lagrangian mechanics and introduce the form of constraint forces in the Lagrangian setting using Lagrange multipliers. A numerical example is considered.

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

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  1. 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. 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. 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. 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. 5 We provide a geometric interpretation of nonholonomic constraints in terms of local constraint surfaces. This allows us to
  6. 6 understand parallel parking as a 'nonlinear' phenomenon and more generally, the problem of accessibility of configuration space when there are constraints.

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