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.