How should I show that the transformation makes the system Hamiltonian?

Math100
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Homework Statement
Consider the system of equations ## \frac{dx}{dt}=(a-by)x(1-x^2), 0<x<1, \frac{dy}{dt}=-(c-dx)y(1-y^2), 0<y<1 ##, where ## a, b, c ## and ## d ## are positive constants.

Show that the transformation ## x=\frac{1}{\sqrt{1+e^{-2q}}}, y=\frac{1}{\sqrt{1+e^{-2p}}} ## makes the system Hamiltonian, with Hamiltonian function ## H(q, p)=ap-b\cdot ln(e^{p}+\sqrt{1+e^{2p}})+cq-d\cdot ln(e^{q}+\sqrt{1+e^{2q}}) ##. (Hint: You may find the following indefinite integral useful: ## \int \frac{dx}{\sqrt{1+e^{-2x}}}=ln(e^{x}+\sqrt{1+e^{2x}}). ##)
Relevant Equations
By definition, a system ## \dot{x}=X(x, y), \dot{y}=Y(x, y) ## is called a Hamiltonian system if there exists a function ## H(x, y) ## such that ## X=\frac{\partial H}{\partial y} ## and ## Y=\frac{\partial H}{\partial x} ##. Then ## H ## is called the Hamiltonian function for the system. A necessary and sufficient condition for ## \dot{x}=X(x, y) ## and ## \dot{y}=Y(x, y) ## to be Hamiltonian is that ## \frac{\partial X}{\partial x}+\frac{\partial Y}{\partial y}=0 ##.
Proof:

Consider the transformation ## x=\frac{1}{\sqrt{1+e^{-2q}}} ## and ## y=\frac{1}{\sqrt{1+e^{-2p}}} ## with the Hamiltonian function ## H(q, p)=ap-b\cdot ln(e^{p}+\sqrt{1+e^{2p}})+cq-d\cdot ln(e^{q}+\sqrt{1+e^{2q}}) ##.
Let ## \dot{x}=\frac{dx}{dt}=(a-by)x(1-x^2)=(a-by)(x-x^3) ## and ## \dot{y}=\frac{dy}{dt}=-(c-dx)y(1-y^2)=-(c-dx)(y-y^3) ##.
By using direct substitution of the transformation ## x=\frac{1}{\sqrt{1+e^{-2q}}} ## and ## y=\frac{1}{\sqrt{1+e^{-2p}}} ##,
we have that ## \dot{x}=\frac{dx}{dt}=(a-\frac{b}{\sqrt{1+e^{-2p}}})(\frac{1}{\sqrt{1+e^{-2q}}}-(\frac{1}{\sqrt{1+e^{-2q}}})^3)=(a-\frac{b}{\sqrt{1+e^{-2p}}})(\frac{e^{q}}{(e^{2q}+1)^{3/2}}) ## and ## \dot{y}=\frac{dy}{dt}=-(c-\frac{d}{\sqrt{1+e^{-2q}}})(\frac{1}{\sqrt{1+e^{-2p}}}-(\frac{1}{\sqrt{1+e^{-2p}}})^3)=-(c-\frac{d}{\sqrt{1+e^{-2q}}})(\frac{e^{p}}{(e^{2p}+1)^{3/2}}) ##.
Note that ## X=\frac{\partial H}{\partial p}=a-b(\frac{1}{e^{p}+\sqrt{1+e^{2p}}})(e^{p}+\frac{e^{2p}}{\sqrt{1+e^{2p}}}) ## and ## Y=\frac{\partial H}{\partial q}=c-d(\frac{1}{e^{q}+\sqrt{1+e^{2q}}})(e^{q}+\frac{e^{2q}}{\sqrt{1+e^{2q}}}) ##.

Above is my proof for this problem. Is everything correct up to here? How should I finish this proof? What needs to be done?
 
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You need to show that \dot q = \frac{\partial H}{\partial p} nad \dot p = - \frac{\partial H}{\partial q} for some H(p,q) (and the question suggests one). Therefore you need to calculate \dot q and \dot p; the easiest way to do that is <br /> \dot q = \frac{dq}{dx} \dot x \qquad \dot p = \frac{dp}{dy}\dot y. Note, however, that x = f(q) and y = f(p) where f(z) = (1 + e^{-2z})^{-1/2}. Rearranging this we have e^{2q} = \frac{x^2}{1 - x^2} so that 2q = 2\ln x - \ln(1 - x^2) and hence \begin{split}<br /> \frac{dq}{dx} &amp;= \frac1x + \frac{x}{1 - x^2} \\<br /> &amp;= \frac{1}{x(1 - x^2)}. \end{split} The denominator neatly cancels the factor of x(1-x^2) in \dot x leaving <br /> \dot q = a - by = a - bf(p). In the same way you can find \dot p.

Now \dot q is a function of p alone and \dot p is a function of q alone, so setting <br /> H = \int \dot q \,dp - \int \dot p \,dq <br /> will give <br /> \dot q = \frac{\partial H}{\partial p} \qquad \dot p = -\frac{\partial H}{\partial q} as required.
 
pasmith said:
You need to show that \dot q = \frac{\partial H}{\partial p} nad \dot p = - \frac{\partial H}{\partial q} for some H(p,q) (and the question suggests one). Therefore you need to calculate \dot q and \dot p; the easiest way to do that is <br /> \dot q = \frac{dq}{dx} \dot x \qquad \dot p = \frac{dp}{dy}\dot y. Note, however, that x = f(q) and y = f(p) where f(z) = (1 + e^{-2z})^{-1/2}. Rearranging this we have e^{2q} = \frac{x^2}{1 - x^2} so that 2q = 2\ln x - \ln(1 - x^2) and hence \begin{split}<br /> \frac{dq}{dx} &amp;= \frac1x + \frac{x}{1 - x^2} \\<br /> &amp;= \frac{1}{x(1 - x^2)}. \end{split} The denominator neatly cancels the factor of x(1-x^2) in \dot x leaving <br /> \dot q = a - by = a - bf(p). In the same way you can find \dot p.

Now \dot q is a function of p alone and \dot p is a function of q alone, so setting <br /> H = \int \dot q \,dp - \int \dot p \,dq<br /> will give <br /> \dot q = \frac{\partial H}{\partial p} \qquad \dot p = -\frac{\partial H}{\partial q} as required.
That was very informative and thorough! I have another question under this same problem.

Show that if ## b>a ## and ## d>c ##, then the transformed system has a centre at ## x=c/d ## and ## y=a/b ##, and prove that in this case, all orbits of the original system are periodic.

My question is, how should I construct a proof for this question? Does the given point ## x=c/d, y=a/b ## indicate a fixed point for this system?
 
Math100 said:
That was very informative and thorough! I have another question under this same problem.

Show that if ## b>a ## and ## d>c ##, then the transformed system has a centre at ## x=c/d ## and ## y=a/b ##, and prove that in this case, all orbits of the original system are periodic.

My question is, how should I construct a proof for this question? Does the given point ## x=c/d, y=a/b ## indicate a fixed point for this system?

Note that the transformation in part (a) has domain (q,p) \in [0,\infty)^2 and range [1,\infty)^2.

How you determine if a point is or is not a fixed point?

If a point is a fixed point, how do you classify it as a centre or otherwise? What are the possible types of trajectory which can occur in a Hamiltonian system?
 
The fixed points occur when ## X(x, y)=Y(x, y)=0 ##.
 
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