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

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SUMMARY

The discussion focuses on the behavior of phase trajectories in a quadratic potential, specifically addressing the upward and downward pointing trajectories. It is established that while the left side of the phase trajectory points downwards, the right side points upwards due to the nature of the potential being unstable. The analysis includes a specific example where a test particle's velocity and position change over time, illustrating that a point with initial conditions of v0<0 and x0>a results in an upward trajectory as the particle accelerates towards the potential.

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  • Understanding of classical mechanics and potential energy concepts
  • Familiarity with phase space and phase trajectories
  • Knowledge of quadratic functions and their graphical representations
  • Basic principles of particle motion in unstable potentials
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  • Study the mathematical formulation of quadratic potentials in classical mechanics
  • Learn about phase space analysis and its applications in dynamical systems
  • Investigate the stability of equilibrium points in potential energy landscapes
  • Explore examples of phase trajectories in different types of potentials, such as harmonic and cubic
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Lambda96
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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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