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.

Total kinetic energy is unchanged between the incoming and outgoing states of an elastic collision. This does not require each body's kinetic energy to be unchanged, nor that kinetic energy remain constant during temporary elastic deformation. In general mechanical energy includes potential energy, so conservation of mechanical energy alone does not imply constant kinetic energy.