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

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In a quadratic potential, phase trajectories can point both upwards and downwards due to the nature of the potential's stability. The left side of the phase trajectory indicates downward motion, reflecting stable conditions, while the right side shows upward motion, which corresponds to unstable conditions where particles accelerate away from the potential minimum. As a test particle moves in this unstable region, its velocity increases, leading to a trajectory that points upwards despite the overall downward acceleration towards the potential. This behavior illustrates the complex dynamics of phase space in relation to potential energy landscapes. Understanding these trajectories is crucial for analyzing motion in phase space effectively.
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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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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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