is there any mathematical proof for law of conservation of energy....

Hi,

As per the law of conservation of energy, energy is neither created nor destroyed, it is only transformed to another type of energy so that the total energy in the universe remains constant.

Let us consider a body undergoing a small displacement, Δx under the action of a conservative force, F.

According to work energy theorem,change in K.E = Work done ⇒ Δk = F(x) Δx … (i)If the force is conservative, the P.E. function, V(x) can be defined such that −ΔV = F (x) ΔxOr, ΔV = − F (x) Δx … (ii)Adding equations (i) and (ii)Δk + ΔV = 0Δ (k + V) = 0, which means(k + V) = Constant.

Thus, total energy remains constant.