Mathematical Definition of Energy

[azabak]

What is the general definition of energy? I already know that it means ability to perform work and that Work = ∫Force d(displacement) = Δ Kinetic Energy = -Δ Potential Energy ( in a conservative field "a closed path integral of the force = 0"), Σ Kinetic-Potential = constant, ∫Kinetic-Potential d(time) = minimum action... so just cut to the chase. None of those concepts define energy in a general mathematical sense. More precisely I'm asking IF there's any definition or not.

 

[Khashishi]

Energy is sort of a bunch of different abstract concepts that all happen to work out to be the same thing. But none of these concepts can really be thought of as the most fundamental or definitive definition of energy. Energy is something we use in physical models--it's not something we measure directly--so any definition is going to seem like a mathematical contrivance or bookkeeping mechanism. The most concrete definition is probably found in relativity. Energy is mass times a unit conversion factor. Energy also can be interpreted as a generator of infinitesimal time displacements.

 

[K^2]

More precisely I'm asking IF there's any definition or not.

Well, what you stated is basically the definition. It's ability to do work. What probably seems confusing about it is that it's hard to define the point where the body can no longer do work. For example, we define gravitational potential energy to be zero at ground level. But what if I dig a hole and lower the object further. It has negative potential energy now. What's that all about?

And the answer to this is that as far as mechanics goes, there is no absolute zero for energy. You can define whatever state of the system you like as zero energy, and then see how much work needs to be done on the system to get it to other states. You call that work the energy of that new state. This is sufficient because in mechanics, only relative energy is important. And so defining energy as ability to work is entirely sufficient as far as all the math goes.

This also holds true for Quantum Mechanics and Thermodynamics. Though, for later, in a really roundabout way, quite often. Where it all starts getting a little crazy is General Relativity. But I wouldn't worry about any of it for now.

 

[azabak]

Although I'm asking this I know one definition of energy. If you define work as W=∫F dr, energy will arise "naturally" as the factor that changes in relation of the speed (kinetic) or the position (potential). Defining both kinetic and potential energy from work leads to its conservation, so that: W = ΔK = -ΔU therefore ΔK+ΔU = 0, Ki-Kf+Ui-Uf = 0, Ki+Ui = Kf+Uf = constant. And since none of those factors are time dependent the constant neither increases nor decreases as time passes, showing an "intuitive" proof the the Noether's Theorem that conservation of energy imply symmetry in time translation. The constant is the total energy, the general definition, but since potential energy is by convention negative the constant is K-U, or the Lagrangian. The closest I can get to define this constant is as the time derivative of the action. The problem is that unless I define action from other quantity (as momentum or angular momentum) this definition will be circular.