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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.

DanielSank
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PP_berry
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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.

Nick Ushor
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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.

Ketan Chaware
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  • When a body is in free fall, energy is conserved at each and every point in its trajectory until it reaches the ground. But gravity is acting on the body. – SchrodingersCat Feb 21 '17 at 13:45
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    @SchrodingersCat Gravity is a internal force.. – Mitchell Feb 21 '17 at 14:27
  • Internal force?? Define your internal force. – SchrodingersCat Feb 21 '17 at 14:30
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    @SchrodingersCat In free fall energy is actually conserved for body-earth system. Otherwise you will not be even able to define potential energy and change in it. So gravity is indeed an internal force as Bhavya Sharma said. – Ketan Chaware Feb 21 '17 at 15:13
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    These points werent there in your answer. You did not speak about conservative forces, said nothing about which system we need to consider, how we define that system. Forces are internal and external only with respect to a particular system and hence we need to define a system to call a force internal or external. Please update your answer. – SchrodingersCat Feb 21 '17 at 15:29
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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 $S$.

gented
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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.

Wouter
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  • You're implying that an environment must stay the same or else energy was not conserved. That is obviously not true. The environment change change a lot while still conserving energy. – JMac Feb 21 '17 at 15:35
  • @JMac I stick to my answer. Note that I didn't say the environment needs to stay the same. I said from the point of view of the system, you cannot see if the environment changes when energy is conserved: the environment cannot have any time-dependent influence on the system. Otherwise time-translational symmetry on the system is lost, whereas conservation of energy is deduced from the Lagrangian/Hamiltonian being not explicitly time-dependent. – Wouter Feb 21 '17 at 15:42