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

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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

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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.

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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:

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

Addendum: Neother's Theorem

Noether's Theorem is one of the most elegant theorems in theoretical physics, named after Emma Noether. It states that "If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time". In this case, conservation of energy can be found using this theorem, whenever we have symmetry in time.

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