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1. Energy Conservation:
If the forces are derived from potential, the total energy \(E=K.E.+P.E.\) is conserved. This can be seen as follows. The equation of motion is \begin{equation}\label{EQ01} m \DD[x]{t} = F(x). \end{equation} If the force is conservative force, it can be derived from a potential \(V(x)\) and we have \begin{equation}\label{EQ02} F(x) = -\dd[V(x)]{x}, \end{equation} and Eq. \eqRef{EQ01} becomes \begin{equation}\label{EQ03} m \DD[x]{t} = - \dd[V(x)]{x}. \end{equation} Multiplying this equation with \(\dd[x]{t}\), we get \begin{eqnarray}\label{EQ04} m \dd[x]{t} \DD[x]{t} &=& - \dd[x]{t} \dd[V(x)]{x}\\ \text{or}\qquad \qquad \frac{1}{2} m \dd{t}\big(\dot{x}^2\big) &=& - \dd[V(t)]{t} \label{EQ05}. \end{eqnarray} This gives \begin{equation}\label{EQ06} \dd{t}\Big(\frac{1}{2}m\dot{x}^2 + V(x)\Big) =0. \end{equation} This equation implies that that the total energy \begin{equation}\label{EQ07} E = \frac{1}{2}m\dot{x}^2 + V(x) \end{equation} is a constant of motion. Energy being a constant of motion means that when the particle moves its position and velocity change from one position to another position, but the value of energy remains constant.
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4727:Diamond Point
