Understanding Fixed Points in Hamiltonian Systems

Master1022
Messages
590
Reaction score
116
Homework Statement
How can we use the potential function of a Hamiltonian system to determine the nature of the equilibrium
Relevant Equations
Hamiltonian system
Hi,

I was attempting a question about Hamiltonian systems from dynamic systems and wanted to ask a question that arose from it.

Homework Question: Given the system below:
\dot x_1 = x_2
\dot x_2 = x_1 - x_1 ^4
(a) Prove that the system is a Hamiltonian function and find the potential function
(b) Use the potential function to determine information about the fixed points of the system (along ## x_2 = 0##)

My question: I don't understand how to do part (b). Specifically, I don't understand why (for potential function ##V##):
"""
- If ##V(\mathbf{x}_{fixed})## is a minimum, the fixed point is a center
- If ##V(\mathbf{x}_{fixed})## is a maximum, the final point is a saddle point
"""

Attempt:
I know that a Hamiltonian system has the form:
\dot x_1 = \frac{\partial H}{\partial x_2} \dot x_2 = - \frac{\partial H}{\partial x_1}

and I can use these relations to calculate the Hamiltonian function:
H(x_1, x_2) = \frac{1}{2} x_2 ^2 + \frac{1}{5} x_1 ^5 - \frac{1}{2} x_1 ^2

By matching terms in ## H(x_1, x_2) = KE + \text{Potential} ##. So I can identify the potential function as: ## V(x_1, x_2) = \frac{1}{5} x_1 ^5 - \frac{1}{2} x_1 ^2 ##

Now in terms of finding the equilibria:
- we can use ##\frac{dV}{dx_1} = 0 ## to find the equilibrium points which end up being: ##(0, 0)##, ##(1, 0)##

Then the solution says:
"""
- If ##V(\mathbf{x}_{fixed})## is a minimum, the fixed point is a center
- If ##V(\mathbf{x}_{fixed})## is a maximum, the fixed point is a saddle point
"""

Where do these come from?/Why is that the case?

Any help would be greatly appreciated.
 
Last edited:
Physics news on Phys.org
If H = \frac12x_2^2 + V(x_1) then the jacobian is <br /> J = \begin{pmatrix}<br /> \frac{\partial \dot x_1}{\partial x_1} &amp; \frac{\partial \dot x_1}{\partial x_2} \\<br /> \frac{\partial \dot x_2}{\partial x_1} &amp; \frac{\partial \dot x_2}{\partial x_2} \end{pmatrix} = <br /> \begin{pmatrix}<br /> 0 &amp; 1 \\<br /> - V&#039;&#039;(x_1) &amp; 0<br /> \end{pmatrix}.<br /> The eigenvalues \lambda of J are then given by <br /> \lambda^2 = -V&#039;&#039;(x_1).
 
pasmith said:
If H = \frac12x_2^2 + V(x_1) then the jacobian is <br /> J = \begin{pmatrix}<br /> \frac{\partial \dot x_1}{\partial x_1} &amp; \frac{\partial \dot x_1}{\partial x_2} \\<br /> \frac{\partial \dot x_2}{\partial x_1} &amp; \frac{\partial \dot x_2}{\partial x_2} \end{pmatrix} =<br /> \begin{pmatrix}<br /> 0 &amp; 1 \\<br /> - V&#039;&#039;(x_1) &amp; 0<br /> \end{pmatrix}.<br /> The eigenvalues \lambda of J are then given by <br /> \lambda^2 = -V&#039;&#039;(x_1).

Many thanks @pasmith ! Ah okay, this is very clear - now I can see the sign of the potential function evaluated at a point will determine whether the eigenvalues are have a non-zero imaginary component or not. Just to check, this rule wouldn't necessarily apply for other Hamiltonian systems because their potential functions may involve both variables and their KE may also have both variables?
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
Back
Top