# Help Understanding Energy in General

• I
• crastinus
In summary: I'm just not sure where to start.In summary, Lagrangian mechanics is a way of describing the behavior of a system by its Lagrangian, which does not change over time. This leads to energy conservation.
crastinus
Consider:

1. Energy is a system's ability to do work.
2. Work is force over a distance.
3. A force is applied over some distance.
4. All physical change (i.e. all change in a system through its phase space) is the result of force.
∴ Therefore, energy would seem to be a system's ability to change. (Here I mean "change" both transitiviely and intransivitiely, both "to change" as into be changed and "to change" as into effect change.)

But that sounds too cute and nice. Or maybe it's just so general that no one would need it.

Any guidance here? Is this reasoning actually flawed? If so, could some more accurate restatement reach the same conclusion? I'm trying to get a sort of root intuitive understanding here, something I've always lacked on the nature of energy.

crastinus said:
Therefore, energy would seem to be a system's ability to change
On the contrary, per Noether's theorem energy is the conserved quantity associated with a Lagrangian that does not change as a function of time.

So, is it better to think of the physical meaning of energy as being closer to a system's ability to stay the same?

The Lagrangian, as I understand it, is (broadly) KE - PE. So, if KE - PE for a system stays at the same value L, then does that mean that the system doesn't change? It would seem not, since it could have the same value L but have different values for KE and PE. I'm thinking of a particle beginning on a trajectory with some velocity and then striking an object and coming to rest. Will the system's L stay the same throughout? But isn't that only because KE and PE change through the transformation?

crastinus said:
Lagrangian, as I understand it, is (broadly) KE - PE. So, if KE - PE for a system stays at the same value L, then does that mean that the system doesn't change?
It is not that the system doesn't change over time, but the Lagrangian for the system doesn't change over time. You can think of the Lagrangian as being the laws of physics for the system. Regardless of whether the system changes or not, it is the fact that the law governing the system does not change which leads to energy.

Also, it is important to understand that we are not talking about the numerical value of the Lagrangian, but its functional form. None of the individual terms can be an explicit function of time.

Interesting!

How is it that the fact that the Lagrangian function for the system doesn't change leads to energy?

I thought I had a sense for that point with respect to the values of L, but you pointed out that that's not what we're talking about here. So, now, I'm not sure of that either.

crastinus said:
How is it that the fact that the Lagrangian function for the system doesn't change leads to energy?
That is what Noether's theorem shows. For every continuous symmetry of the Lagrangian there is a corresponding conserved quantity.

Translational symmetry (laws of physics are the same here and there) leads to conservation of momentum.

Rotational symmetry (laws of physics are the same facing this direction and that direction) leads to conservation of angular momentum.

Time translation symmetry (laws of physics are the same yesterday and today) leads to conservation of energy.

Thanks for the help. I'm not sure I quite "get it" yet.

I guess I'm just thinking of energy as something a thing has. And so I'm trying to see what it is that the thing has when it has energy. It made some intuitive sense to me that a system that has energy can change, whereas one with no energy cannot.

That seems to be either wrong or just not relevant. I have the sense that this is one of those basic things, through the failure to understand which I am held back.

crastinus said:
I guess I'm just thinking of energy as something a thing has. And so I'm trying to see what it is that the thing has when it has energy
I can't help you with this other than to recommend that you abandon this view of energy. It is not a material or even a property that a thing has or contains.

Are you familiar with the Lagrangian approach to physics?

Somewhat familiar in that I've read some about Lagrangian mechanics, but, honestly, I never felt that I actually understood it. For example, the action S as the sum of the Lagrangian from time(a) to time (b). I can solve the equation, even show that δS=0, but I don't know what the physical meaning is.

Thanks for the patience with this. I am a classicist actually (you know, Ancient Greek and Latin and all that) trying to study physics for the joy of it. I love it and am profoundly interested, but I think some of the basic ideas of the more advanced stuff just escapes me yet.

I understand. Sometimes the Lagrangian feels like voodoo. You just write it down, turn the mathematical machinery on, and out pops the answer.

But the insights you get from the mathematical framework are important. Noether's theorem, in particular, is one of the most pivotal results in physics.

crastinus said:
Consider:

1. Energy is a system's ability to do work.
2. Work is force over a distance.
3. A force is applied over some distance.
4. All physical change (i.e. all change in a system through its phase space) is the result of force.
∴ Therefore, energy would seem to be a system's ability to change. (Here I mean "change" both transitiviely and intransivitiely, both "to change" as into be changed and "to change" as into effect change.)

But that sounds too cute and nice. Or maybe it's just so general that no one would need it.

Any guidance here? Is this reasoning actually flawed? If so, could some more accurate restatement reach the same conclusion? I'm trying to get a sort of root intuitive understanding here, something I've always lacked on the nature of energy.

Words on their own can be confusing. It may help to look at the equations which use energy and then try to state what they mean in words.

I would start with mechanics. There are different forms of energy. Understand the difference between potential energy and kinetic energy. Know how the unit of energy (Joule) is derived from the basic units. Work some problems involving energy. This can all be done using just algebra and some simple trigonometry.

Once the use of energy in mechanics is understood, look at energy in electromagnetic theory. Later you can learn about quantum mechanics and relativity which will provide more insights into the nature of energy.

It is precisely solving energy problems that suggested to me that energy of the various types is a property of systems.

Here's my original line of reasoning, modified:

1. All transformations of physical systems require energy of one kind of another.
2. Without energy of any kind, a system could not undergo transformations of any kind. (Indeed, for physics, it would seem that a system with no energy is not a system at all.)
∴ So, energy in general seems to be a physical system's capacity for transformation.

I can see what the various kinds of energy are supposed to be, more or less. I'm trying to see what they have in common and therefore what it is that they are kinds *of*, i.e. what energy is in general. (It's true that an answer here probably won't help us solve physics problems better, but it does seem interesting.)

crastinus said:
4. All physical change (i.e. all change in a system through its phase space) is the result of force.
This is related to what @Dale said; per Newton's first law, you don't need to know anything about an object's motion history to know its speed now and therefore its current kinetic energy. As such, a definition that utilizes force is extraneous and too restrictive.

Energy is a property of an object's current state and not something that requires tracking "change".
I guess I'm just thinking of energy as something a thing has.
I agree: it is a property.
And so I'm trying to see what it is that the thing has when it has energy. It made some intuitive sense to me that a system that has energy can change, whereas one with no energy cannot.
It is more generic than that: energy can cause "change" but can also be a product of "change".

For example, when you burn something flammable, you use its own internal chemical energy to affect a change. But potential energy can be changed by applying a force to make an object move, so in that way the change in energy is the result of whatever happened.

crastinus said:
It is precisely solving energy problems that suggested to me that energy of the various types is a property of systems
I tend to think of "properties" as being quantities that depend only on the system, and not on the coordinates. Energy is different in different reference frames, so if it is a "property" then being a "property" doesn't imply a lot of the things that I would expect it to imply.

Dale said:
I tend to think of "properties" as being quantities that depend only on the system, and not on the coordinates. Energy is different in different reference frames, so if it is a "property" then being a "property" doesn't imply a lot of the things that I would expect it to imply.
I used both words above, but bolded the more correct one: energy is a state
https://en.wikipedia.org/wiki/State_function

It is a function of certain properties, but I would agree that it is not really a property of an object, but you might say it is a property of a system.

Dale
russ_watters said:
It is a function of certain properties, but I would agree that it is not really a property of an object, but you might say it is a property of a system.
I probably have called it a property sometime in the past too. I don't think that it is necessarily wrong, but I am concerned that @crastinus wants to attach some materialistic connotation to the term that isn't warranted.

russ_watters
@Dale I'm content with the minimal idea of a property as a quantity that depends only on the system.

@russ_watters I think I see what you mean. My phrase "the capacity for transformation" can have two meanings: in one sense it is the capacity to bring about a transformation; in another it is the capacity to undergo a transformation. It seems that you are saying that both are included in what energy is in general. That seems right to me.

My larger question would then be: Does defining the energy of a physical system in general as the capacity for transformation (in both senses) either have counterexamples or seem to refer also to things beside energy?

Dale said:
I probably have called it a property sometime in the past too. I don't think that it is necessarily wrong, but I am concerned that @crastinus wants to attach some materialistic connotation to the term that isn't warranted.

You may not be wrong either! ;)

But, for me, thinking of energy as the capacity for transformafion is what makes clear that it is *not* anything material, but rather the result of the way systems themselves are (or something like that). Does that make sense?

crastinus said:
@Dale I'm content with the minimal idea of a property as a quantity that depends only on the system
Then energy is not a property because it also depends on the coordinates as well as the system. The same system described using different coordinates will have different energy.

crastinus said:
Does defining the energy of a physical system in general as the capacity for transformation (in both senses) either have counterexamples or seem to refer also to things beside energy?
Yes. If you have a time varying Lagrangian then you have change without a defined energy. This is the case in cosmology where the universe does not have a defined energy but is clearly changing. So you can have change without energy.

Similarly, a system which does not change would have a Lagrangian which is trivially time invariant. It would therefore have a conserved energy. So you can have energy without change.

russ_watters said:
I used both words above, but bolded the more correct one: energy is a state
https://en.wikipedia.org/wiki/State_function

It is a function of certain properties, but I would agree that it is not really a property of an object, but you might say it is a property of a system.

OK! State function. So, energy is not one property a system has (like position or charge), but a state of the system's properties. (I guess I would still call that a property in some sense, but I see your point; technically, it's not.)

Does that sound like I'm understanding it more correctly?

Dale said:
Then energy is not a property because it also depends on the coordinates as well as the system. The same system described using different coordinates will have different energy.

Yes. If you have a time varying Lagrangian then you have change without a defined energy. This is the case in cosmology where the universe does not have a defined energy but is clearly changing. So you can have change without energy.

Similarly, a system which does not change would have a Lagrangian which is trivially time invariant. It would therefore have a conserved energy. So you can have energy without change.

But is change without a defined energy the same as change without energy at all? Would one describe such a Lagrangian as having no energy? Would it be right to say that such a Lagrangian has no energy state function?

As to energy without change, I would just suggest that what's of interest here is not change as such, but energy as the capacity for change. Can there be a Lagrangian for a system not just that has no energy state function but for a system that has no capacity to have an energy state function?

crastinus said:
But is change without a defined energy the same as change without energy at all?
If I don't define "flubnubitz" then all statements using it are equally meaningless. Such as: "Is change without a defined flubnubitz the same as change without flubnubitz at all?"

crastinus said:
Can there be a Lagrangian for a system not just that has no energy state function but for a system that has no capacity to have an energy state function?
I can't think of a way to interpret this question that makes it answerable.

In the end, I don't think that this line of thought is valuable. It appears to be directly contradicted by the usual Lagrangian formulation of physics. If it even could be tweaked to become correct, it seems that it would take more mental effort to do so than it would just to learn the standard approach.

Dale said:
If I don't define "flubnubitz" then all statements using it are equally meaningless. Such as: "Is change without a defined flubnubitz the same as change without flubnubitz at all?"

I can't think of a way to interpret this question that makes it answerable.

In the end, I don't think that this line of thought is valuable. It appears to be directly contradicted by the usual Lagrangian formulation of physics. If it even could be tweaked to become correct, it seems that it would take more mental effort to do so than it would just to learn the standard approach.

Fair enough.

I am obviously new at this kind of thinking, so go easy on me! :) If I said anything obnoxious, I'm sorry.

And thanks for the help, by the way. It's really something that strangers are willing to help a wayward Greek professor who feels like he's stumbling around in the dark. Ask me about Middle Egyptian or the history of logic, and I'll be a researcher with a clue. But about energy? Not so much, apparently.

## 1. What is energy?

Energy is the ability to do work or cause change. It can take many forms, such as thermal, electrical, chemical, and mechanical.

## 2. How is energy measured?

Energy is measured in joules (J) or kilowatt-hours (kWh). Joules are used for smaller units of energy, while kilowatt-hours are used for larger amounts.

## 3. What is the difference between potential and kinetic energy?

Potential energy is stored energy that an object has due to its position or condition, while kinetic energy is the energy an object has due to its motion. For example, a ball sitting at the top of a hill has potential energy, while a ball rolling down the hill has kinetic energy.

## 4. What are some examples of renewable and non-renewable energy sources?

Renewable energy sources include solar, wind, hydro, geothermal, and biomass energy. Non-renewable energy sources include fossil fuels like coal, oil, and natural gas.

## 5. How is energy converted from one form to another?

Energy can be converted from one form to another through various processes such as combustion, nuclear reactions, and electromagnetic induction. For example, a power plant converts the kinetic energy of falling water into electrical energy through hydroelectricity.

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