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.