# Why definition of potential energy and law of conservation of mechanical energy is misleading several times?

I regularly see 1 or 2 questions on this website about the definition or application of potential energy.The users fundamentally ask the same thing in every question.

What I have learned till now is:-$$dU_{system}=-dW_{int,cons}$$

The change in potential energy of the system is defined as the negative of work done by the internal conservative forces of the system

What I have read is that it is defined only for a multi particle system and absolute potential energy is not defined yet.We have only defined relative potential energy.

The definition is quite obvious because if we choose a single particle system then no forces would be internal and we can't define potential energy corresponding to it.I have a rigid intuition about the term negative in the definition.

Why people on this website continuously argue with me that potential energy is also defined for a single particle system?It seems vague to say that Potential energy of this particle is 10 joules.

Now coming to law of conservation of mechanical energy of a system of particles:-

For any system of particles we have from work energy theorem:-$$dW_{total}=dK_{system}$$ $$dW_{int,con}+dW_{int,non-con}+dW_{ext}=dK_{system}$$ $$-dW_{int,con}=dU_{system}$$ $$dW_{int,non-con}+dW_{ext}=dU_{system}+dK_{system}$$ $$dU_{system}+dK_{system}=dE_{mech,system}$$ $$dE_{mech,system}=dW_{int,non-con}+dW_{ext}$$

In context of this law,people argue with me about the term of work done by external forces on this system.According to whatever I have learned till now is that this law is also valid for a multi particle system.

I want to ask here about the correct and generalized definition of potential energy as well as the correct law of conservation of mechanical energy,if I am wrong?

The arguments are in the following question Question on the definition of the potential energy for a two particle system

• In which questions are the arguments you mention? – G. Smith Jul 7 at 4:46
• I think your argument is correct, but I would be using $\Delta$ rather than $d$ because $d$ implies an exact differential, which non-conservative work is not. In fact work is generally not an exact differential, and is generally not written with a difference symbol. I wonder about including non-conservative work in the definition of mechanical energy. I have not seen in included, but perhaps it is conventionally included. If not, we would have $W = \Delta E_\mathrm{mechanical} + \Delta E_\mathrm{non-cons}$. For clarity, the last term can be omitted. – garyp Jul 7 at 5:21
• By the way, you might get voted to close. However, PSE does allow for someone to post an "educational" question of general interest, and the poster can provide an answer. If you start to get complaints about this question you could delete it, and then as a new question "What are the definitions of potential energy and mechanical energy?" and then answer it. If competent people don't like your answer they will tell you, and you could correct it as needed. – garyp Jul 7 at 5:32
• Are you concerned about the definitions given in the books or people's understanding or lack thereof? – user207455 Jul 7 at 5:44