# Force as gradient of scalar potential energy

If a particle is acted upon by the forces which are conservative; that is, if the forces are derivable from a scalar potential energy function in manner $F=-\nabla V$.

I was just wondering what may be the criteria for force to be expressed as negative gradient of scalar potential energy and HOW DO WE PROVE IT?

Your Question all but includes the right search term for an Answer from Wikipedia, "Conservative Forces", which gets you to http://en.wikipedia.org/wiki/Conservative_Forces. There's even what you ask for, a proof. There's also another link to http://en.wikipedia.org/wiki/Conservative_vector_field, which gives some quite good visualizations that will probably help. Loosely, there mustn't be any vortices in the force field for there to be a scalar potential energy that generates the force vector field as $\nabla\!\cdot\!\phi(x)$.

The force field indeed must be conservative, and this is the criterion for your ability to express force in terms of potential energy. To test this let a force field (in 2D for simplicity) be given by $$F=f_i+g_j$$ where $f$ and $g$ are functions of $x$, $y$ then the partial differential of $f$ with respect to $y$ must be equal to the partial differential of $g$ with respect to $x$. This is a general result in mathematics, to test for conservative vector fields.

In the case of a simple Newtonian gravitational field, $$f=GmMx/(x^2+y^2)^{3/2}$$

Integrating with respect to $x$ yields $-GmM/(x^2+y^2)^{1/2}+C(y)$, where $C(y)$ is the constant of integration. Differentiating this result with respect to y yields $GmMy/(x^2+y^2)^{3/2}+dC(y)/dy$. Equating this to $g$ let us compute $dC(y)/dy=0$ and therefore $C(y)=C$, where $C$ is a constant independent of $x$ or $y$.

Hence the potential energy is given by $$Ep=-GmM/(x^2+y^2)^{1/2}+C$$

The minus sign is simply an outcome of arbitrary definitions, the acceleration of gravity vector is defined to point down (toward the earth's center of mass), positive distance is defined in the upward direction (or increasing away from the earth's center of mass), we define the work done as an increase in potential energy, positive work done is force times distance ; so the mass times a negative downward g times a positive upwards distance needs a minus sign to increase the potential energy.

That is the definition of a force. In my opinion, we assume Energy, Space, Momentum and Time as fundamental and build the theory of mechanics based on these quantities.

• No, that is not the definition of force. Only certain forces can be represented as the gradient of a potential. That's what the question was about. – Mark Eichenlaub Oct 30 '11 at 10:51
• That's not the definition of force, but modifying it the answer seems fine. It is just we built the idea of force from experience, and also of energy. Then OP expression is postulated for conservative forces. As leonar susskind call it: ' potential energy principle'. – user153036 Oct 17 '18 at 16:40