Conservation of mechanical energy ($T+V=\mathrm{constant}$)

For constant masses, a time-independent potential energy $V$ and forces ${\mathbf{F}}_i=-{\nabla }_{i}V+{\mathbf{R}}_i$ with zero total extra power ${\sum }_i{\mathbf{R}}_i\cdot {\mathbf{v}}_i=0$, Newton's second law gives $dT/dt=-dV/dt$. Thus mechanical energy is conserved. Smooth stationary constraint reactions and a purely magnetic Lorentz force supply zero power. This is more restrictive than conservation of energy in general, which may include thermal or other internal forms.