Understanding Potential Energy and Functions: Verification and Clarification

[cmmcnamara]

Hi all,

So I do feel a bit silly with this question, but I've just begun to realize the relation ship between potential energy and potential functions...

So just quickly, would the following relationship be considered true:

$$\frac{d\vec{p}}{dt}=\nabla\phi$$

and by breaking apart the vectors, generally:

$$\frac{d\vec{p_s}}{dt}-\frac{d\phi}{ds}=0$$

Just wanted a quick verification and/or push in the right direction.

Thanks!

 

[Muphrid]

Usually there's a negative sign in the first equation, because we define potential energy as decreasing along the direction of force. Otherwise, your logic is correct.

 

[cmmcnamara]

I'm not sure I understand the negative sign, wouldn't that be dependent on the orientation of the force (eg, if my net force is pointed upwards from earth, potential energy increases with height).

 

[Muphrid]

The gradient gives the direction of increase of a scalar field.

When the scalar field is potential energy, the force is always defined as the direction that potential energy decreases. Hence, a minus sign is required.

 

[PJC01]

Hi there,

Don't feel silly at all! It's great that you're taking the time to understand potential energy and potential functions. To answer your question, yes, the relationship you have mentioned is indeed true. In general, potential energy is the energy possessed by a system by virtue of its position or configuration. It is often represented by the symbol "U" and is a scalar quantity. On the other hand, potential functions are mathematical functions that describe the potential energy of a system in terms of its position or configuration. These functions are often represented by the symbol "φ" and are also scalar quantities. Therefore, the relationship between the two can be expressed as the gradient of the potential function being equal to the change in momentum of the system over time. This can be written as \frac{d\vec{p}}{dt}=\nabla\phi.

Breaking down the vectors, as you have mentioned, can help in understanding this relationship better. The change in momentum of a system can be represented as \frac{d\vec{p_s}}{dt}, where \vec{p_s} is the system's momentum. On the other hand, the change in potential energy over distance can be represented as \frac{d\phi}{ds}, where "s" represents the distance. Therefore, the relationship can also be written as \frac{d\vec{p_s}}{dt}-\frac{d\phi}{ds}=0, showing that the change in momentum and the change in potential energy over distance are equal and opposite, resulting in a constant total energy for the system.

I hope this helps clarify your understanding of potential energy and potential functions. Keep up the great work!