# Conditions for Conservation of energy law

In two-dimensional motion, which conditions are needed to be satisfied so the conservation of energy law holds? (for example, simple pendulum motion)

• Energy is always conserved. Maybe you mean to ask "when is mechanical energy conserved"? – udiboy1209 Jul 29 '13 at 15:58
• Yes, I meant that. :) – gov Jul 29 '13 at 16:01

Mechanical energy is conserved when there are no non-conservative forces acting on the body. Examples are friction and elastic forces of stress in a body. These non-conservative forces convert mechanical energy to other forms of energy like heat and sound. So the mechanical energy is not conserved, while the total energy is.

• So, for simple pendulum, how can I show that total mechanical energy is conserved? I know that forces acting on a bob are gravitational force and the one on the string (I don't know what's the name of that force; maybe straining force, I'm not sure). – gov Jul 29 '13 at 16:16
• @Gorica, The force in the string is called tension force. Both gravity and tension are conservative forces. You should read about conservative forces on wikipedia – udiboy1209 Jul 29 '13 at 16:19

Whenever the Lagrangian of the system doesn't explicitly depend on time; there is a conserved quantity which we call it energy.

$$\frac{dE}{dt}=0 \Leftrightarrow \frac{\partial L}{\partial t}=0$$

In this context, $E$ is defined as:

$$E = \sum_i p_i\dot{q_i}-L$$