משפט עבודה אנרגיה ושימור אנרגיה

dW=\vec{F}\cdot\vec{dx}=m\vec{a}\cdot\vec{v}dt=m\vec{v}\cdot\left(\vec{a}dt\right)=m\vec{v}\cdot\vec{dv}=d\left(\frac{1}{2}mv^{2}\right)

\Rightarrow W=\Delta\left(\frac{1}{2}mv^{2}\right)=\Delta E_{k}

U\left(\vec{x}\right)=-\int^{\vec{x}}_{\vec{x_{0}}}\vec{F}\left(x\right)\cdot\vec{dx}

E_{k}\left(\vec{x}_{2}\right)-E_{k}\left(\vec{x}_{1}\right)=W=\int^{\vec{x}_{2}}_{\vec{x_{1}}}\vec{F}\left(x\right)\cdot\vec{dx}=\int^{\vec{x}_{0}}_{\vec{x_{1}}}\vec{F}\left(x\right)\cdot\vec{dx}+\int^{\vec{x}_{2}}_{\vec{x_{0}}}\vec{F}\left(x\right)\cdot\vec{dx}=U\left(\vec{x}_{1}\right)-U\left(\vec{x}_{2}\right)

E_{k}\left(\vec{x}_{1}\right)+U\left(\vec{x}_{1}\right)=E_{k}\left(\vec{x}_{2}\right)+U\left(\vec{x}_{2}\right)=\const

d\left(E_{k}+U\right)=dE_{k}+dU=dE_{k}-dW=0

E_{k}+U=\const

m\ddot{x}-f\left(x\right)=0

0=m\ddot{x}\dot{x}-\dot{x}f\left(x\right)=\frac{d}{dt}\left(\frac{1}{2}m\dot{x}^{2}+U\left(x\right)\right)

\frac{1}{2}m\dot{x}^{2}+U\left(x\right)=\const