Math 260 Conservative systems

Save your work regularly--you never know when Mathematica may crash.  

Email your completed file to tleise@amherst.edu by Monday 4pm.

Plotting commands, with double well example

Plot[-x^2/2+x^4/4,{x,-2,2},PlotLabel->"Potential function V(x)"]

Plot3D[y^2/2-x^2/2+x^4/4,{x,-1.5,1.5},{y,-1.5,1.5},ViewPoint->{-0.881, -3.263, -0.161},PlotRange->{-1/4,1/2},PlotLabel->"Energy surface z=E(x,y)"]

ContourPlot[{y^2/2-x^2/2+x^4/4==0,y^2/2-x^2/2+x^4/4==1/4,y^2/2-x^2/2+x^4/4==1/2,y^2/2-x^2/2+x^4/4==-1/8,y^2/2-x^2/2+x^4/4==-1/5},{x,-2,2},{y,-2,2},PlotLabel->"Trajectories are contours of energy surface"]

Exercises

Consider the system (Overscript[x, ¨])=x-(x^2).

Exercise 1.   Find and plot the potential function V(x), noting where the max and min values occur.
Exercise 2.   Rewrite the DE as a system in the xy-plane.
Exercise 3.   Find and classify the fixed points.
Exercise 4.   Find a conserved quantity E(x,y) using the total energy formula (kinetic plus potential--what must m be?).
Exercise 5.   Plot the surface given by this conserved quantity.  Use Plot3D and make sure the domain contains the fixed points.
Exercise 6.   Where does the energy surface have a local minimum? What does that tell you about the fixed point corresponding to that local minimum?
Exercise 7.   Find an equation describing the trajectories that start and/or end at the saddle point, which also separate the closed and nonclosed trajectories and which includes a homoclinic orbit (closed trajectory that begins and ends at a saddle point).
Exercise 8.   Plot the homoclinic orbit plus some closed and nonclosed trajectories, using ContourPlot.