Conservative System

--Example--

Duffing Eqn

Lin Eqn

eq = m*x''[t] + k*x[t] + k1*x[t]^3

With an Applied Conservative Force

feq  = eq + D[v[x], x]

Find Constant Quantities

(feq2 = Expand[D[x[t], t]*(feq)]) == 0

Notice that {dv \over dx}*{dx \over dt} = {d \over dt} V(x(t)) by chain rule.

(feq3  =  
          feq2 /. v'[x] x'[t] -> D[v[x[t]], t]) == 
      0

(eqConst  = Integrate[feq3, t]) == Constant

What might the physical interpretation of this constant be?

Solve[eqConst  ==  Const,  v[x[t]]]

1. find behavior with different forcing functions -- see example 6.5.2 page 160
2. Solve for the homogenous case.  F[x] ==0 so v[x] ==0.

Example Solution (to option 2)

{m, k, k1} = {1, 1, 0}

energy[x_, 
        y_] := eqConst /. v[x[t]] -> 0 /. 
          x[t] -> x /. x'[t] -> y

energy[x, y]

Plot3D[energy[x, y], {x, -1, 1}, {y, -1, 1}]

ContourPlot[energy[x, y], {x, -1, 1}, {y, -1, 1}]

Book Example Pg 161

energy2[x_, y_] := 1/2 y^2  -  1/2 x^2 + 1/4 x^4

Plot3D[energy2[x, y], {x, -2, 2}, {y, -2, 2}, 
      ViewPoint -> {1.398,  4.387,  0.718}, 
      BoxRatios -> {1, 1, 1}]

ContourPlot[energy2[x, y], {x, -2, 2}, {y, -2, 2}]