Conservation of energy

Energy is conserved in a closed system. In dynamics problems, we often link the principle of conservation of energy with work, where $U$ is work, $T$ is kinetic energy, and $V$ is potential energy.

Our full work-energy equation is:

$$T_1+V_1+U_{1\cdot 2}=T_2+V_2$$

Only considering kinetic energy:

$$T_1+U_{1\cdot 2}=T_2$$

If we consider all forces other than gravitational forces and spring forces:

$$U_{1\cdot 2}=\Delta T+\Delta V$$

If the only forces are gravitational, elastic, and nonworking constraint forces, then the work term is zero and the energy equation becomes:

$$T_1+V_1=T_2+V_2$$ $$E_1=E_2$$