- #1
Master1022
- 611
- 117
- 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:
[tex] \dot x_1 = x_2 [/tex]
[tex] \dot x_2 = x_1 - x_1 ^4 [/tex]
(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:
[itex] \dot x_1 = \frac{\partial H}{\partial x_2} [/itex] [itex] \dot x_2 = - \frac{\partial H}{\partial x_1} [/itex]
and I can use these relations to calculate the Hamiltonian function:
[tex] H(x_1, x_2) = \frac{1}{2} x_2 ^2 + \frac{1}{5} x_1 ^5 - \frac{1}{2} x_1 ^2 [/tex]
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.
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:
[tex] \dot x_1 = x_2 [/tex]
[tex] \dot x_2 = x_1 - x_1 ^4 [/tex]
(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:
[itex] \dot x_1 = \frac{\partial H}{\partial x_2} [/itex] [itex] \dot x_2 = - \frac{\partial H}{\partial x_1} [/itex]
and I can use these relations to calculate the Hamiltonian function:
[tex] H(x_1, x_2) = \frac{1}{2} x_2 ^2 + \frac{1}{5} x_1 ^5 - \frac{1}{2} x_1 ^2 [/tex]
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: