One Dimensional Equations of Motion

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The general Lagrangian for one degree of freedom which we call $x$ is \begin{align} L=\frac{m \cdot \dot{x}^2}{2}-U(x) \end{align} It is possible to integrate this equation in general form by using energy conservation. \begin{align} \frac{m \cdot \dot{x}^2}{2}+U(x)=E \tag{1} \end{align} Rewriting (1) as \begin{align} \frac{dx}{dt}=\sqrt{\frac{2}{m} [E-U(x)]} \end{align} we can integrate by seperating variables \begin{align} t=\sqrt{\frac{m}{2}} \int{\frac{dx}{\sqrt{ E-U(x)}}} + const. \end{align}


If $U(x)<E$ then motion is calls infinite otherwise it's called finite. To make this more clear, we study the example with the potential $U(x)$ shown above. At some points $x$ the potential is larger than the energy. This means if our initial conditions lie between $x_1$ and $x_2$, the particle will oscillate between A and B and will not be able to overcome the hill from B to C. As the energy of the system is increased B and C move closer to one another and vanish for $U(x)<E$. Then the particle is able to overcome the potential. A possible solution would be, that the particle comes from large $x$, goes to point A where it is reflected and moves to $x \rightarrow \infty$.