Lecture 11 - Tue 2023 10 17
Phase plane.
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From Lecture 10, finish/do:
Rabbits and Sheep example.
Nonlinear pendulum.
Dissipative nonlinear pendulum [Note energy here becomes a Liapunov function].
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From Lectures 9-11 Matt Durey notes. % ------------------- % Reversible systems.
I will only sketch the main points in the lecture; for details read the notes.
Define what reversible means.
Motivate by example of mass in potential: \ddot{x} + V^\prime(x) = 0.
Point out that reversible and conservative are not the same.
Example of a conservative system that is not reversible: left to a problem set.
Example of a reversible system that is not conservative:
  Given by drawing phase plane.
  --- Place two saddles along the x-axis, at some x = \pm a.
  --- Place two nodes   along the y-axis, at some y = \pm b.
      Node at y = b unstable and node at y = -b stable.
  --- Connect repelling main axis of node at y = b with attracting direction
      of the saddles, and make the repelling secondary axis the y axis.
  --- Reflect across x-axis and reverse arrows.

More examples in Lectures 9-11 Matt Durey notes [read by students]
Example bottom page 10: Reversible system
     \dot{x} =  y-y^3   and   \dot{y} = -x-y^2.
Example bottom page 11: Reversible/Non. Conservative system.
     \dot{x} = -2*cos(x)-cos(y)   and   \dot{y} = -2*cos(y)-cos(x).
     Reversible across line x = -y.    x to -y; y to -x; t to -t;
     Note system has a symmetry across x = y;  but it does not flip time
     [so, not related to reversibility ... need the time reversal for the
      argument that linear centers are also nonlinear centers].
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                                                                           SKIP
We have seen that a system can be reversible but not conservative.
Can a system be conservative, but not reversible?
    Yes: look at  \dot{x} = E_y   and   \dot{y} = -E_x    [A]
    
         and select E with level curves near a local minimum that have no
         symmetry.

By the way: [A] belongs to a very important class of conservative systems:
            "Hamiltonian systems".
            All physical, non-dissipative systems are Hamiltonian!
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