# Sign of Hamiltonians: Plotting U & Phase Portraits

• luitzen
In summary, a Hamiltonian is a mathematical function used in classical mechanics to describe the total energy of a physical system. Plotting U, or the potential energy, in Hamiltonian systems allows us to visualize the energy landscape and understand the behavior of the system. To plot a phase portrait for a Hamiltonian system, we need to find the equations of motion using the Hamiltonian function and then use a computer program or graphing software. A phase portrait can provide information about the system's stability, direction and speed of motion, and presence of periodic or chaotic behavior. Phase portraits have various real-world applications, such as analyzing spacecraft stability and predicting chemical reaction behavior.
luitzen
How do you decide on the sign of a Hamiltonian function?

For example, I have the following system of differential equations:

$x'=y$
$y'=-\dfrac{3}{2}x^{2}-2x$

With the following Hamiltonian:
$$H^{\oplus}=\dfrac{1}{2}x^{3}+x^{2}+\dfrac{1}{2}y^{2}$$

because $\dfrac{dH^{\oplus}}{dt}=0$. But if $\dfrac{dH^{\oplus}}{dt}=0$ then $\dfrac{dH^{\ominus}}{dt}=0$ is also true.

We can write $H=U\left(x\right)+T\left(v\right)$ with $v=y$. We can then use $U\left(x\right)$ to construct the phase portrait of the system of differential equations.

With the use of Matlab I created the phase portrait of the system and it is obvious that the Hamiltonian with the positive sign leads to the correct plot. My question now is how do I know which Hamiltonian I should use?

Plot of $U^{\oplus}$:

Plot of phase portrait:

Last edited:
The sign of the Hamiltonian function is determined by the nature of the plot. In this case, you can see that the plot of the phase portrait is consistent with a Hamiltonian with a positive sign. This indicates that H^{\oplus} is the correct choice in this situation.

## 1. What is a Hamiltonian?

A Hamiltonian is a mathematical function used in classical mechanics that describes the total energy of a physical system. It takes into account both the kinetic energy and potential energy of the system.

## 2. What is the significance of plotting U in Hamiltonian systems?

Plotting U, or the potential energy, in Hamiltonian systems allows us to visualize the energy landscape of the system. This can help us understand the behavior of the system and identify any stable or unstable equilibrium points.

## 3. How do you plot a phase portrait for a Hamiltonian system?

To plot a phase portrait for a Hamiltonian system, we first need to find the equations of motion using the Hamiltonian function. Then, we can use a computer program or graphing software to plot the phase portrait, which shows the trajectories of the system over time.

## 4. What information can we gather from a phase portrait?

A phase portrait can provide information about the behavior of a Hamiltonian system, such as the stability of equilibrium points, the direction and speed of motion, and the presence of periodic or chaotic behavior.

## 5. How can we use phase portraits in real-world applications?

Phase portraits can be used in a variety of real-world applications, such as analyzing the stability of a spacecraft or predicting the behavior of a chemical reaction. They can also help us understand complex systems and make predictions about their behavior.

• Differential Equations
Replies
6
Views
1K
• Differential Equations
Replies
2
Views
2K
• Differential Equations
Replies
16
Views
1K
• Differential Equations
Replies
1
Views
1K
• Differential Equations
Replies
3
Views
2K
• Differential Equations
Replies
2
Views
1K
• Differential Equations
Replies
4
Views
984
• Differential Equations
Replies
2
Views
816
• Calculus and Beyond Homework Help
Replies
1
Views
387
• Differential Equations
Replies
5
Views
1K