- #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.

[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##)

"""

- 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

"""

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

"""

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: