Why is potential energy negative?

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?

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

jtbell
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.

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 $\frac{1}{2}mv^{2}-mgh=L$.