Conceptual derivation of (classical mechanical) energy?

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I realized that I was unable to bridge the gap in a satisfactory way between F=ma and W=Fd or T=1/2mv^2.
[...]

So you see in my view energy is just an artificial number that happens to be conserved in addition to momentum conservation, since we only have inverse square law forces.

Without inverse square laws, maybe our world wouldn't conserve energy.

(I haven't generalized my proof to non-instantaneous forces yet :( )

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Is energy just a mathematical fiction / convenience, like imaginary numbers (OK - perhaps I opened a can of worms with that statement :-) or is there some physical "reality" to it?
[...]

Hi Roy,

I believe I did the very thing you requested: I bridged from F=ma to W=Fd, using just d=1/2*a*t² and v=a*t. That was in post #24 of this thread. I'm very curious what you think about that demonstration.

Cleonis

Well, angular momentum [tex] m r^2 \omega [/tex] is a quadratic function of 'r', but there does not seem to be a way of interpreting that as akin to a metric.
I agree that interpreting kinetic energy as related to a metric is tempting, but I think it's a dead end.

The challenge is, I suppose, to capture in words the most general sense of the concept of Energy.

It appears that materials have a state of zero energy, and that energy is accumulated when the material is pushed away from this ground state.

When an elastic material is deformed it stores energy, which is released when the material relaxes again. In elastic deformation the molecules of the material do not slide along each other (when they do slide you have plastic deformation, which dissipates energy). In the case of elastic deformation you are deforming the very molecules away from their ground state.

In a molecular bond there is a distance between the atoms that is the state of least energy. I suppose the most fundamental level of description we currently have is the following: given the quantum physics of molecular orbitals there is a state (of the molecule) that is the most probable. When a molecule is deformed away from that most probable state then it tends to return to the most probable state.

Another aspect: is it possible to view gravitational potential energy as a deformation away from a state of lowest energy? I think so. According to GR gravitational interaction is mediated by deformation of spacetime. (In other words: spacetime curvature acts as mediator of gravitational interaction.) I infer that when two objects are pulled apart then there is a net increase in spacetime deformation.

Finally, kinetic energy.
When two objects have a velocity relative to each other then that two-object system has the potential to do work.
Example, a electric car that is designed for regenerative braking. When the car and the Earth have a velocity relative to each other then regenerative braking will recharge the batteries. The state of lowest energy is where the two objects have no relative velocity.

So in all I like to think of the relative velocity of the two objects in a two-object system as a form of potential energy.
I don't so much think of the quadratic form, I think of the potential to do work, expressed as force acting over a certain distance.
What is special of course is that in the case of kinetic energy this potential is realized only at the very instant in time that the two objects actually interact. When two objects interact with each other, and not with other surrounding objects (as in a collision) then they are in effect a two-object system.

In the case of nuclear physics the nuclear binding energy makes a difference. The mass of a nucleus is not just the sum of the constituent protons and neutrons, the internal energy has inertial mass too.

In the case of a very large nucleus, such as a Uranium nucleus, there is so much repulsive force between the protons that the nucleus is teetering on the brink of falling apart. A neutron absorbtion event can trigger nuclear fission. The combined mass of the two fission products is smaller than the mass of the original Uranium nucleus.

In a sense the Uranium nucleus is under very large internal stress. There is the strong nuclear force that acts to keep the nucleus together, and the Coulomb repulsion between the protons acts to make the nucleus fission. This internal stress is a form of potential energy, and it has a corresponding inertial mass. The magnitude of that inertial mass is given by the following expression:

[tex] m = \frac{E}{c^2} [/tex]

Roy,

I find the most successful learning method is recursive. I call it a spiral. Only (a coherent)part of a subject is presented on each circuit of the centre. As we move outwards the diameter expands and our knowledge expands and more is presented.

This is why the sort of treatise prepared by a professor is often poor (=heavy going) teaching material but an excellent reference once the subject has been mastered.

I commend you to the little monograph written by M B Glauert, entitled Principles of Dynamics.

This monograph is pitched at about the level you seem to require, and contains many explanatory insights, in particular his explanation of what is meant by a particle when introducing Newton's laws.

[...] this step is unnecessary. Just solve (2) for t and substitute into (1) and you can get to (4) pretty quickly. [...]

[...] In this view, energy is the more fundamental concept, and forces are just derived from it. [...]

@Riad
There's a counterexample. Just imagine 3 charged masses aligned (named 1, 2 and 3. 2 is the central one). Suppose charge of 1 = charge of 3, and distance 1-2 = distance 2-3. Than 1 will exert in 2 a force, but only 1 will be displaced.

You don't need to impose that those point masses have the same mass nor that they are equal. Just use Newton's definition F=dp/dt, and use Newton's third law in order to find that, regardless of the properties of thoses masses, or the quantity of bodies under study, we will always have conservation of momentum.

In fact, you should start backwars: conservation of momentum always holds (even in relativity or quantum mechanics), but there are situations in which Newton's third law doesn't apply. So you can't really use Newton's third law to find cons. of momentum, but use cons. of momentum from Newton's third law.

Cleonis
I think what you're saying is that to really appreciate energy, we have to go beyond the simplification of our Newtonian worldview and embrace relativity.

I have a pedagogical question, and a philosophical question, both involving energy:

What exactly is energy, anyway?
Direct answers or pointers to relevant references would be appreciated.

Thanks!
Roy

Actually no, that's kind of a step away from what I intended.

The question you raised was whether the concept of Energy can be seen as essentially a bookkeeping device, useful as a tool, but that we shouldn't attribute physical existence to it.

Well, the lesson from relativistic physics is that to Matter and Energy we have to attribute the same level of existence; they're on equal par, given that both matter and energy have inertial mass.

The nuclear binding energy example is, I guess, the only example where we can actually measure the mass difference that arises from difference in internal energy.
Theoretically a compressed spring has additional inertial mass, corresponding to the stored energy, but that mass difference is far, far too small to be measurable.

I think to really appreciate what energy is, you have to go beyond the approximation that is Newtonian physics and turn to relativity. From that vantage point, one can see that matter and energy are on equal par, as you say. Under Newton, matter and energy are two very different things, and one must view energy as essentially a fallout of the math under that approximation to "reality".