Does an analogous idea of energy exist and satisfy conservation?

[0rthodontist]

Under what general formal mathematical conditions does an analogous idea of energy exist and satisfy conservation? In a gradient vector field, energy as force times distance exists and is conserved in some sense. What is a more general idea? What, conceptually, makes energy "energy"? Is energy just any conserved quantity or is there another essential characteristic?

 

[Nimz]

There are other physical quantities that are conserved for the same mathematical reasons as energy, like total angular momentum. However, angular momentum isn't energy - it doesn't even have the same units. As I recall, the difference between energy and angular momentum is in the choice of coordinates when using the lagrangian. Energy comes from using rectangular coordinates, angular momentum comes from using spherical coordinates. If your generalized space has units other than distance, you can come up with other conserved quantities when using the lagrangian.

So energy isn't just any conserved quantity. It is a specific conserved quantity. You might want to ask the question of what the other essential characteristics are in the physics section.

 

[Astronuc]

Well first of all energy is a scalar quantity and momentum (like velocity) is a vector quantity.

In physics, energy is considered the 'ability to do work', and it can be kinetic (motion) or potential. See - http://hyperphysics.phy-astr.gsu.edu/hbase/enecon.html

Momentum - tp://hyperphysics.phy-astr.gsu.edu/hbase/mom.html#mom

Conservation laws - http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html#cons

Mathematics is simply a systematic method of describing the relationships observed in physics - mathematics is more or less the language of physics.

 

[Curious3141]

Under what general formal mathematical conditions does an analogous idea of energy exist and satisfy conservation? In a gradient vector field, energy as force times distance exists and is conserved in some sense. What is a more general idea? What, conceptually, makes energy "energy"? Is energy just any conserved quantity or is there another essential characteristic?

I think the concept of "energy" has a specific physical meaning. But if you're looking for mathematical formalism surrounding the concept, I'd suggest Lagrangian and Hamiltonian Mechanics.

 

[0rthodontist]

I'm not looking for specific physics. Energy is a very broad concept even in physics that includes not only Newtonian mechanics as you linked to, but also every other physical formulation. In Newtonian mechanics, if my understanding is correct, energy is conserved because all force fields involved are gradients. Is there a similarly simple generalized reason in modern physics?

My interest is that energy is a useful tool for analyzing physical systems. So a generalization of energy might be a useful tool for analyzing some mathematical or computational systems. Entropy has been generalized via math to apply to algorithms and such, and energy is somewhat related to entropy, so perhaps there is a like generalization.

If a system is given as a set of objects and states for those objects, together with a next-state function, I think a quantity called energy would have to have the following characteristics. It would have to be defined for a given next-state function and computed from any group of objects and states under that function. It would have to be invariant for all groups of objects and states under applications of the function. And it would have to depend sensitively on the objects and states of the system, in the sense that for any object of the system, removing the object, adding a new object, or changing the state of the object in an appropriate manner will change the energy of the system.

I think may be something to be said in this context about conversion of energy from one form to another, but I am unsure.

Mathematics is simply a systematic method of describing the relationships observed in physics - mathematics is more or less the language of physics.

This is not intended as a philosophical discussion, but I take the opposite view: physics is an instance of mathematics.

 

[HallsofIvy]

"Energy" is NOT a mathematical concept- "conservation" of a quantity is. One can argue that the history of the concept of energy is a continuing struggle to HAVE a conserved quantity! It was relatively late that it was recognized that heat is a form of energy- and that "definition" of heat as energy was precisely to maintain conservation of energy. And, of course, with the recognition, in relativity, that mass itself could be converted to energy, we had to accept mass as a type of energy, simply to maintain "conservation of energy".

 

[0rthodontist]

The fact that physicists have been so successful at maintaining conservation of energy suggests that energy is fairly fundamental. This is in contrast to, for example, conservation of mass. Maybe energy has no analogue in systems that differ much from the universe, but why assume that? Energy absolutely can be formulated mathematically in our particular universe. What I'm looking for is not energy per se--that's just a casual use of the word. I am looking for energy-like quantities.

 

[Nimz]

Would you be interested in something like knot energies, then? Knot energy isn't energy in the physical sense, but would seem to have some of the properties you are looking for. I don't know much about the subject of knot energy (yet), so I can't help you with any details.

 

[matt grime]

Energy absolutely can be formulated mathematically in our particular universe.

But is not *defined* mathematically. You are asking this in completely the wrong (sub)forum. It is just, mathematically, some number, that is invariant under some conditions. There are literally hundreds of numerical invariants in mathematics, which is thankfully far richer than Astronuc's descriptions of it would have people believe. (I'm equally sure he'd not like me describing physics as merely one small application of mathematics, which is neither true nor accurate.)