2. The function W(x, y) has a local minimum at (0,1) and a saddle at (0, —1). The level curves of...

Question

2. The function W(x, y) has a local minimum at (0,1) and a saddle at (0, —1). The level curves of W(x, y) are illustrated below.
(i)
0)
x
Write down the set of first order Hamiltonian ODE's (in terms of W(x, y)) for which W(x, y) is the Hamiltonian. Draw the phase plane for this system and indicate the steady states and their nature. You do not need to indicate the direction of flow in this case. (ii) Write down the first order gradient system of two ODE's for which W(x, y) is the gradient function. Draw the phase plane for this system and indicate the steady states and their nature. You do need to indicate the direction of flow in this case.

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Robert · Accepted Answer

Answer (i) 
For a Hamiltonian Function W(x,y) we can write the two ODE’s as following 
�̇�? = ????(??,??)
????
                 ----------------------------1 
�̇�? = -  ????(??,??)
????

2. The function W(x, y) has a local minimum at (0,1) and a saddle at (0, —1). The level curves of W(x, y) are illustrated below. (i) 0) x Write down the set of first order Hamiltonian ODE's (in terms...

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