Lecture 10 - Thu 2023 10 12
Phase plane.
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From Lecture 9, finish "Example: 1-D mass with double well potential"
                and "The standard tricks to get conserved quantities."
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From Lectures 9-11 Matt Durey notes. % ---------------------------------------- %
I will only sketch the main points in the lecture; for details read the notes.

Example on top of page 4.  \dot{x} = -x*(1-x)*(1+x)  and \dot{y} = -2*y
     Saddle, node, saddle on real axis.
     Special property: nullclines are also "special" solutions. UNUSUAL!
     Compute Jacobians and special orbits on edge of quadrant
             [x, y cannot go neg.]
     Get phase portrait.

Example on top of page 6. % ------------------------------------- TO LECTURE 11 %
Rabbits and Sheep [this is also in the book].
     \dot{x} = 3*x - x^2 - 2*xy (rabbits)
     \dot{y} = 2*y - y^2 - x*y  (sheep)       
               Note: logistic terms + competition.
               Rabbits have edge on growth; sheep on competition.
     Compute fixed points and Jacobian, then classify.
     Get phase portrait.  Point out the "gap" in argument --- to be filled
                          by index theory and P. Bendix.
     Note prediction: "competitive exclusion" results [model too simplistic].

Example bottom of page 8.  \ddot{x} + V^\prime = 0;  V = -0.5*x^2 + 0.25*x^4.
                           Double well potential.

Example bottom of page 9.  The importance of being ISOLATED.
     \dot{x} = x*y  and \dot{y} = -x^2  has E = x^2+y^2 conserved.
     Draw phase portrait.
     What happened here? Do not get closed orbits near min./max.
     "Good energy" is as follows:
     1) Fixed points are isolated.
     2) grad E \neq 0 at non-critical points.
     3) grad E = 0 at critical points.
     4) The hessian of E is non-degenerate at the C.P., so surface nearby
        is either a saddle or a local max./min.
     Energies without this properties are useful, but you need to
     analyze them separately.
     THEN: max/min of E yield centers and saddles yield saddles.
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Unplanned extras % ----------------------- % Lagrangians and Variational Forms.

Started with questions such as:
1) Given a dynamical system, how do I find what the orbits do?
   There is no "universal scheme". Like many other things, you learn and gain
   intuition by practice, and build a tool box. When you face a new system,
   maybe the tool box is enough, or maybe not. Then you need to develop new
   tools, which is may not be possible.
2) Given a system, how do you find if it has a conserved quantity?
   Basically, same answer as for (1). An interesting thing here is that now a
   day, theoretical physicists propose new models starting from the need to
   conserve certain things. Using Lagrangian formulations guarantees this, see
   below. But if you have a system and want to know if it has a Lagrangian,
   this is hard.
   Introduce Lagrangians: First history "nature wants to minimize certain
   things", example: Fermat's principle. Modern view: at the fundamental level
   systems follow extrema of the action = \int dt L.
   Show how equations follow from this, with simple example L = L(x, \dot{x}).
   Analogous to calculus; the "gradient" of the action must vanish ... but
   here the action is a function of the trajectory [a curve in phase space].
   So, gradient gets replaced by variational derivative.
   Mention Noether's theorem: If action invariant under (a) time translation:
   conservation of energy; (b) space translation: conservation of linear
   momentum; (c) rotation: conservation of angular momentum; etc. [other
   invariances are possible].
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% ------------------------------------------------------- BELOW: To Lecture 11 %
Example on page 12. Nonlinear pendulum; Use energy to draw phase plane.

Example on page 13. Dissipative nonlinear pendulum. % ------------------------- %
     Small dissipation. Use the equation for the energy [point out
     dissipation] to construct phase diagram.
     Here the energy gives us a first peek at Liapunov functions.

pp 10-11. Reversible systems [if time permits, else next lecture] % ----------- %
     Important: centers preserved IF critical point is on the symmetry axis.
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