Take x ¨ = w − 2 x + x 2 , where w ≥ 0. Show that the evolution of x conserves a form of the energy and identify the potential function, sketching the phase portrait for w = 0

Question

Take             x      ¨        =  w  −  2  x  +      x    2  , where   w  ≥  0. Show that the evolution of x conserves a form of the energy and identify the potential function, sketching the phase portrait for   w  =  0

Teagan Zamora · Accepted Answer

In physics courses we tend to use the following trick: multiply             x      ¨        =  w  −  2  x  +      x    2   by             x      ˙       to obtain            x      ˙                  x      ¨        =  (  w  −  2  x  +      x    2    )            x      ˙              ⇒            d          d      t            (          1      2                                x          ˙                    2        )    =      d          d      t            (    w    x    −          x      2        +          1      3              x      3        )  Consequently,      d          d      t            (          1      2                                x          ˙                    2        −    w    x    +          x      2        −          1      3              x      3        )    =  0from which we find  E  =      1    2                      x        ˙              2    −  w  x  +      x    2    −      1    3        x    3  is a constant of the motion. Call it energy if you like.

Take ddot(x)=w-2x+x^2, where w >= 0. Show that the evolution of x conserves a form of the energy and identify the potential function, sketching the phase portrait for w=0

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