[NOTES/ME-06001]- Application of Energy Conservation Law

EOM from Energy Conservation

In the previous section law of energy conservation was derived from EOM. In one dimension, the energy conservation law leads to EOM by reversing the steps from Eq07 to Eq02. Thus differentiating Eq07 w.r.t. gives \begin{equation}\label{EQ08} m \DD[x]{t} \dd[x]{t} + \dd[V(x)]{x}\dd[x]{t} =0. \end{equation} This implies that \begin{equation}\label{EQ09} m \DD[x]{t} + \dd[V(x)]{x} =0 \Longrightarrow m \DD[x]{t} =- \dd[V(x)]{x} . \end{equation} which is just the equation of motion.

Solving the EOM

Using energy conservation law it is straight forward to solve the time dependence of position. We get the time dependence of position, \(x(t)\), by first solving for velocity \(\dot{x}\), and integrating the resulting equation w.r.t. time. We will write out the steps now. \EqRef{EQ06} gives \begin{eqnarray}\nonumber \dot{x} &=&\sqrt{\frac{2}{m}(E-V(x))} \nonumber \Rightarrow dt &=& \frac{dx}{\sqrt{\frac{2}{m}({E-V(x)})}} \\ \Rightarrow t-t_0 &=& \int_{x_0}^x\frac{dx}{\sqrt{\frac{2}{m}{(E-V(x))}}}. \label{EQ10} \end{eqnarray} This equation gives \(t\) as function of position. Solving this result for \(x(t)\) will give the position as a function of time \(t\). Thus the problem of solving the equation of motion and obtaining the position as function of time reduces to quadratures, i.e. to evaluation of integrals.

• Note that the equation of motion is a second order differential equation in time. On the other hand the energy conservation law leads to a first order differential equation for \(\dot{x(t)}\). So the conservation law leads to a simpler differential equation.