# Why is potential energy negative?

Hertz
Total energy is $\frac{1}{2}mv^{2}-mgh=C$ for a classical non-rotating object right? C can be determined based on an arbitrary initial condition that suits the problem, right? How come it's not $\frac{1}{2}mv^{2}+mgh=C$? This seems like a more suitable definition of "total" energy. I'm sure there's a reason it's minus and not plus, but what is the reason? Does it have something to do with the direction of the conservative force field that the object is in?

## Answers and Replies

Mentor
It just depends on the definition of h. If h is upwards, it is +mgh, otherwise it is -mgh.

Mentor
Total energy is $\frac{1}{2}mv^{2}-mgh=C$ for a classical non-rotating object right?

According to who? That's not what I see in the textbooks that I've used, which all have a + sign, and define h as increasing in the upwards direction.

chandra.phys
Total energy is $\frac{1}{2}mv^{2}-mgh=C$ for a classical non-rotating object right?

If you have taken the surface of the earth as the reference point for your potential energy, then the potential energy of a body of mass m at a height h (small compared to earth's radius) is mgh. It is a positive quantity with respect to the reference point. The total energy in such a case is $\frac{1}{2}mv^{2}+mgh$. The Lagrangian can be $\frac{1}{2}mv^{2}-mgh=L$.