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
223
75
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

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

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

Phase trajectories point upwards and downwards in a quadratic potential because the system's energy is conserved, and the quadratic potential represents a parabolic shape where the kinetic and potential energies exchange as the system evolves. The trajectories reflect the oscillatory nature of the motion, with velocity and position changing in a manner that traces out these characteristic shapes in phase space.

What is a quadratic potential?

A quadratic potential is a type of potential energy function that is proportional to the square of the displacement from an equilibrium position. Mathematically, it can be represented as V(x) = (1/2)kx^2, where k is a constant and x is the displacement. This form of potential is typical for harmonic oscillators, such as springs and pendulums under small oscillations.

How do phase trajectories relate to energy in a quadratic potential?

In a quadratic potential, the total energy of the system is the sum of kinetic and potential energies. Phase trajectories in this context represent contours of constant energy. As the system evolves, the energy remains constant, leading to closed or elliptical trajectories in phase space that reflect the conservation of energy and the harmonic nature of the motion.

What does the shape of the phase trajectory indicate about the motion of the system?

The shape of the phase trajectory in a quadratic potential indicates periodic motion. For a simple harmonic oscillator, the phase trajectory is an ellipse, showing that the system undergoes sinusoidal oscillations. The upward and downward directions in the phase plot correspond to the system moving through its equilibrium position with varying velocity.

Can phase trajectories in a quadratic potential ever be straight lines?

No, phase trajectories in a quadratic potential cannot be straight lines. This is because the motion in a quadratic potential is inherently oscillatory, leading to curved trajectories in phase space. Straight lines would imply non-oscillatory, unbounded motion, which is not characteristic of systems governed by a quadratic potential.

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