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Potential Energy

It can be shown mathematically that if a force F is such that the integral of F . d l vanishes around any closed-path in space, i.e. if

then the components of F are derivatives of a (scalar) function U (x, y, z) as given below:

[The Proof: According to Stokes theorem in vector calculus, we have,

for an arbitrary surface area A enclosed by the closed-path. Hence,

Since curl of a gradient is always zero. Minus sign is taken in the definition for the reason which will become clear below.]

That is, a conservative force can be written as,

F = F_x i + F_y j + F_z k

where the symbol U is called gradient of U. The symbol represents a vector differential operator whose components in Cartesian co-ordinates are . Note that U is not an explicit function of t: U = U (x, y, z) only.

Now, for a force F given, we find that,

= U_a – U_b

where, U_a = U(x_a, y_a, z_a) and U_b = U (x_b, y_b, z_b). That is, the work done depends only on the end points a and b and not on the path followed.

Note that dW = -dU. The function U has the dimensions of energy and one therefore calls U (x, y, z) as the potential energy of the particle at position (x, y, z). By definition, work done by conservative force in moving the particle from position a to b to negative of the potential energy change:

Combining above with the work-energy theorem, we find

The above equation represents the conservation of energy principle for conservative forces. It says that the sum of kinetic energy of the particle (or an object) remains constant during motion under conservative forces. In the world of mechanics, it is sometimes called law of conservation of energy.