Conservation of Energy: Proving the Relation ((mdx^2)/(dt^2))=F(x)

[belleamie]

A) Given ((mdx^2)/(dt^2))=F(x)
prove conservation of energy

 

[quasar987]

Bonjour mon amie. Vous êtes belle aujourd'hui.

Here's how roughly: Define K(x) as $K(x) = \frac{1}{2}mv(x)^2$. Define $W_{a \rightarrow b}$ as $\int_a^b F(x')dx'$. Show that $W_{a \rightarrow b} = K(b) - K(a)$. Then define U(x) as $-\int_s^x F(x')dx'$ where s is a fixed arbitrary point, and show that $U(b) - U(a) = -W_{a \rightarrow b}$. Then define the energy E of the system as E(x) = K(x) + U(x). Show how the previously found result imply that E is a constant.