Single particle problem

The single particle problem deals with the simplest case of a physical interaction of a particle with a force field . The field is usually given by a potential , so that the potential energy only depends on the location of the particle. It is assumed that the field exists independently of the particle and is not influenced by the movement of the particle. In one dimension , the one-particle problem with the energy law can be solved by a simple integration and subsequent inversion : ${\displaystyle V(\mathbf {x} )}$

${\displaystyle {\begin{aligned}{\frac {1}{2}}m{\dot {x}}^{2}+V(x)&=E\\\Leftrightarrow {\frac {dx}{dt}}&={\sqrt {{\frac {2}{m}}\left[E-V(x)\right]}}\\\Rightarrow \int _{t_{0}}^{t}{\frac {dx}{\sqrt {{\frac {2}{m}}\left[E-V(x)\right]}}}&=t-t_{0}\end{aligned}}}$

This trick can be used in higher dimensions if there are further symmetries and the resulting conservation quantities, such as the angular momentum in the central potential , see also the Kepler problem .