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My text book reads

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?

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3 Answers 3

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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 $\nabla\!\cdot\!\phi(x)$.

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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 $$Ep=-GmM/(x^2+y^2)^{1/2}+C$$

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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.

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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

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