Why Is Force Defined as the Negative Gradient of Potential Energy?

[lightswitch]

F_x= (-dU)/dx

He used dn in place of dx in a different example.

My professor wrote this on the board, and I think he tried to explain why but I didn't get it. Normally he's pretty good, but I'm not understanding the relationship here. Why is this true?

 

[Nabeshin]

Do you know anything about Lagrangian mechanics?

 

[lightswitch]

I do not.

 

[Matterwave]

You don't need to know Lagrangian mechanics to understand this.

This is essentially a definition.

You define the potential energy U of a conservative force F at two different points in space by the amount of work needed to move an object from one location to another at constant velocity.

Thus, the potential energy U is defined as a line integral of F (since work is force times distance).

The result with the derivative is simply an application of the fundamental law of calculus.

Of course, you could also define it the other way around if you wanted, but this integral definition tends to be more intuitive.