1. a. Consider the Duffing oscillator equation with no forcing y ‘ = v v ‘ = y – y 3 . What are…

1.       a.    Consider the Duffing oscillator equation with no forcing

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1. a. Consider the Duffing oscillator equation with no forcing y ‘ = v v ‘ = y – y 3 . What are…
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y ' = v

 

v ' = y y3.

 

What are the equilibrium points for this system?

 

b.          Show that this system is Hamiltonian with Hamiltonian function

 

H(y, v) = v2/2 – y2/2 + y4/4.

 

c.           Sketch the level curves of this Hamiltonian function.

 

d.          Describe the motion of the slender steel beam in the different cases that arise.