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What is the formal definition of energy in physics?

My question is what is the definition of energy in physics in general ? Moreover is energy a naturally occurring quantity or is a term defined in theoretical physics, ( for example force is somewhat a naturally occurring and not a defined quantity, whereas work is defined term)?

Qmechanic
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Bijayan Ray
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  • @ACuriousMind I went across the different question but I want a definition in mathematics sense in theoretical physics sense not in a naive philosophical sense, so those answers do not clarify my doubts. – Bijayan Ray Mar 31 '19 at 14:24
  • (At least some of ) The answers to the questions I linked clearly state that energy is the Noether charge of time translation. How is that not a mathematical definition? – ACuriousMind Mar 31 '19 at 14:25
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    @BijayanRay There is no formal definition of energy in the mathematical sense you're looking for. This is physics, not math. The Noether charge 'definition' ACuriousMind talked about above is as rigorous as you can get. – Avantgarde Mar 31 '19 at 14:52
  • @Avantgarde So do you mean energy is a naturally occurring quantity like force, then what are the laws governing the behaviour of energy ? – Bijayan Ray Mar 31 '19 at 15:01
  • What do you mean by a naturally occurring quantity? It is as physical as the concept of force (I would argue it is more physical because Newtonian physics is messy). – Filipe Miguel Mar 31 '19 at 15:29
  • One doubt I have from the naive definition of energy in physics is energy in Introductory physics is often defined as ability to do work $$$$ offcourse it is intuitive but does not it requires an extra law to support that energy in a situation is same that is say for example a ball is in the wall on one side we have a air medium and another side a viscous fluid, what makes us to conclude in whatever direction we roll the ball on both side the work done by the ball for or against gravity and viscous force will be equal? $$$$ I would like to have this apparently simply doubt to be clarified? – Bijayan Ray Mar 31 '19 at 15:40
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    @BijayanRay Energy as the ability to do work is a poor man's definition. No one uses that definition beyond high school. – Avantgarde Mar 31 '19 at 16:36
  • @Avantgarde Then what definition for energy is used in general? – Bijayan Ray Mar 31 '19 at 16:45
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    @BijayanRay What ACuriousMind wrote above. – Avantgarde Mar 31 '19 at 17:12
  • Thanks to all of you for your efforts and time . – Bijayan Ray Mar 31 '19 at 17:46
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    "Energy as the ability to do work is a poor man's definition. No one uses that definition beyond high school." Nonsense. That was the underlying definition of energy for over 200 years. It wasn't until Emmy Noether that this definition was replaced by something actually better. – dmckee --- ex-moderator kitten Mar 31 '19 at 18:54
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    @dmckee That's what I meant. So how is that nonsense, exactly? I don't see anyone using that old 'definition' in a random arxiv paper today. – Avantgarde Mar 31 '19 at 22:55
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    @Avantgarde That is the definition of work that is used in upper division and (at least many) graduate mechanics texts. Oh, know one says "how can this system do work?" because we bootstrap the "ability to do work" definition through the work-energy theorem and the infrastructure of potential energies and what not, but that definition is where all that stuff came from. To dismiss a seminal idea like that as "high school" shows a lack of understanding of just how far that notion took us. Hamiltonian mechanics presages Noether in a way and it came from the old way of thinking about energy. – dmckee --- ex-moderator kitten Apr 01 '19 at 19:08
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    @dmckee The question isn't about the understanding of concepts. It's about a formal definition. – Avantgarde Apr 01 '19 at 20:47

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