Exact definition of law of conservation of energy

Question

I have learnt about law of conservation of energy overtime that "For an isolated system, energy of that system will remain constant with time." I want to know the conditions and constraints under which this law works. Basically i want the exact definition of law of conservation of energy with all the conditions in which it can be applied.

Answer 1

The condition and constraint is in the title. The system needs to be isolated. If it is not isolated, the law obviously isnt applicable to that system. A system is isolated when it is so far removed from (all) other systems that it doesnt interact in any way with them. So where could you find these isolated systems? You'll find them in theory, because strictly and ideally isolated systems do not actually occur in experiments or in nature.

Answer 2

Consider any closed isolated system. There is no external agent doing work on system (that's what isolated means!). Then energy in such systems is conserved. The only condition is that there should be no net external force doing any work. But there can be any number of internal forces like gravitation, electrostatic etc working on different components of system.

If you take relativity into consideration, then extra condition is that mass of system should remain constant.

Answer 3

Let $S$ be a mechanical system that allows a Lagrangian description in terms of $L(q, v, t)$: if such Lagrangian does not explicitly depend on time, namely if $ \partial{L}/\partial t = 0 $ then one has that neither does the quantity $$ E_{L}= v\frac{\partial{L}}{\partial v} - L(q,v,t). $$ The latter is referred to as energy of the mechanical system $S$.

Answer 4

First of all, you have to draw a line between your system and its environment.

Imagine now the system is replaced by e.g. you. If, from that viewpoint, you could in any way see that time is passing by, energy is not conserved. If everything in the environment you can see remains the same, energy is conserved.