Why do phase trajectories point upwards and downwards in a quadratic potential?

In summary, the conversation discussed the concept of motion in phase space and introduced a quadratic potential at minute 4. At minute 6, the phase trajectory was introduced and pictures were provided for better understanding. The individual expressed confusion about the direction of the phase trajectory on the right side, as the potential is unstable and should lead to a downward acceleration. However, it was explained that the trajectory on the right side can also point upwards due to the nature of the potential and the trajectory being "pointed" upwards at certain points in time.
  • #1
Lambda96
158
59
Hi,

I am currently preparing for my exam and have just watched a video about motion in phase space.



From minute 4 a quadratic potential is introduced and then from minute 6 minute the phase trajectory.

Here are the pictures

quadratic potential
Bildschirmfoto 2023-02-02 um 12.41.54.png


phase trajectory
Bildschirmfoto 2023-02-02 um 12.42.28.png


Regarding phase trajectory on the left side, I understand that these are pointing downwards, but I don't understand why the phase trajectory on the right side is pointing upwards, I would say that these must also be pointing downwards, as the potential is unstable and no matter where I place a test particle, it will always accelerate downwards towards the potential.
 
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  • #2
Take a point on the curve with v0<0, x0>a. At the next instant in time, 0>v1>v0 and a<x1<x0 so the curve is ”pointed“ upwards.
 
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Likes Lambda96

1. Why do phase trajectories point upwards and downwards in a quadratic potential?

Phase trajectories point upwards and downwards in a quadratic potential because of the nature of the potential function. In a quadratic potential, the force acting on an object is directly proportional to the displacement from the equilibrium point. This results in a restoring force that causes the object to oscillate around the equilibrium point, resulting in phase trajectories pointing upwards and downwards.

2. How does the shape of the potential affect the direction of phase trajectories?

The shape of the potential directly affects the direction of phase trajectories. In a quadratic potential, the potential function has a parabolic shape, which results in a linear force. This linear force causes the object to oscillate around the equilibrium point, resulting in phase trajectories pointing upwards and downwards.

3. Can phase trajectories point in other directions in a quadratic potential?

No, phase trajectories can only point upwards and downwards in a quadratic potential. This is because the linear force from the parabolic potential function only allows for oscillations in the vertical direction. Any other direction would require a non-linear force, which is not present in a quadratic potential.

4. What is the significance of phase trajectories in a quadratic potential?

Phase trajectories in a quadratic potential are significant because they represent the motion of an object in the potential field. By analyzing the shape and direction of the phase trajectories, we can understand the behavior and stability of the system. They also provide a visual representation of the energy and momentum of the object in the potential field.

5. How do phase trajectories change with different initial conditions in a quadratic potential?

The initial conditions, such as the initial position and velocity, determine the specific phase trajectory of an object in a quadratic potential. Changing these initial conditions will result in a different phase trajectory, but it will still follow the general pattern of oscillating upwards and downwards. The amplitude and frequency of the oscillations may vary depending on the initial conditions.

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