Understanding Noether's Theorem: Conservation of Momentum and Energy

[ubavontuba]

The laws of momentum and conservation state that you can't accelerate/move the center of mass for an isolated system off of its center of gravity without applying an external force, correct?

If you could do it with an internal force, this would therefore be a conservation of momentum violation, right?

If you used an energy source from within the system to accomplish this, would it violate the conservation of energy laws too (assuming you converted the energy directly into motion and/or displacement of the center of gravity)?

Note: I'm not stating or attempting to imply that any of this is possible, I'm just trying to better understand Noether's Theorem.

 

[Tide]

First, can you explain how this relates to Noether's Theorem?

 

[Integral]

Have you ever played with a gyroscope?

 

[ubavontuba]

This is a test, this is only a test...

First, can you explain how this relates to Noether's Theorem?

The short version of Noether's Theorem goes something along the lines of this:

Emmy Noether proved that symmetrical systems are conserved, and conserved systems are symmetrical. Also, force, momentum and energy are conserved in isolated (symmetrical) systems, and they are conserved separately.

Is that close?

As I understand it, in terms of the isolated system I've presented above, it means that for my system to act as described it must be inherently asymmetrical. (Note: As far as I know, asymmetry in isolated systems has not been observed.)

I've seen a lot of proposals for "sci-fi" drives that require no reaction mass, often based on asymmetrical fields and/or space-time curvatures caused by some sort of internal generator. It seems to me that if these were to work, then the conservation laws would need to be re-examined.

If energy could be directly converted into motion without expelling mass/energy, wouldn't that mean also that the universe must have some sort of coordinate system that identifies an energy potential for various locations as being different for here, versus over there?

If not, then if something like this really worked, then wouldn't that mean that energy could be destroyed (lost?)?

The CP(T) violation points to the possibility that symmetry and conservation don't necessarily hold for large scale (universe wide) isolated systems, so I'm wondering what such an event on a local scale might be like and what the rammifications for it might be.

P.S. If the energy used to accomplish the movement directly increased the temperature of the system in proportion to the energy used and the motion was only temporary (no permanent momentum change), would that mean that conservation still holds? (Sort of like reaction wheel physics)

 

[ubavontuba]

Spin-dizzy

Have you ever played with a gyroscope?

Integral, you are too funny! (I suppose that you're suggesting that my question regards "gyroscopic propulsion.")

I might be cracked, but I'm not that cracked.

Wait! Now I get it! That's why flying saucers are shaped like that! They're large gyroscopes!

Ha-ha-ha-ha-ha!

 

[Integral]

Integral, you are too funny! (I suppose that you're suggesting that my question regards "gyroscopic propulsion.")

I might be cracked, but I'm not that cracked.

Wait! Now I get it! That's why flying saucers are shaped like that! They're large gyroscopes!

Ha-ha-ha-ha-ha!

I am not sure what you are laughing about?

When the center of a gyroscope is spinning you can consider it an energy source. When you try to rotate the the spin axis of the center you can feel the energy transfer to the outside world.

 

[ubavontuba]

I am not sure what you are laughing about?

I (mistakenly?) thought you were alluding to the crackpot notion of gyroscopic (propellantless) propulsion and that you were trying to insinuate that that is where I was headed with this thread. I apologize for misunderstanding.

When the center of a gyroscope is spinning you can consider it an energy source. When you try to rotate the the spin axis of the center you can feel the energy transfer to the outside world.

Oka-a-a-ay, but what has that to do with my original questions? I'm not following you here. Please elaborate.

 

[Tide]

assuming you converted the energy directly into motion and/or displacement of the center of gravity

I have no idea how you would accomplish that but, at the very least, if you are going to apply Nother's Theorem to such a problem you're going to have to come up with a Lagrangian or Hamiltonian description of it.

 

[ubavontuba]

I have no idea how you would accomplish that but, at the very least, if you are going to apply Nother's Theorem to such a problem you're going to have to come up with a Lagrangian or Hamiltonian description of it.

If you can't (or won't) even try to help me answer my questions, why are you even bothering to post here? Are you purposefully intending to be mean to me?

 

[topsquark]

The laws of momentum and conservation state that you can't accelerate/move the center of mass for an isolated system off of its center of gravity without applying an external force, correct?

If you could do it with an internal force, this would therefore be a conservation of momentum violation, right?

If you used an energy source from within the system to accomplish this, would it violate the conservation of energy laws too (assuming you converted the energy directly into motion and/or displacement of the center of gravity)?

Note: I'm not stating or attempting to imply that any of this is possible, I'm just trying to better understand Noether's Theorem.

I'm not sure if this answers your question but consider the following (Freshman Physics book question, Serway I think, but a lot of them probably have it):

A cannon and a pile of cannon balls are placed on one side of an otherwise empty railroad car. One by one the cannon balls are fired at the opposite side of the railroad car. What happens to the railroad car?

To quickly sum up what happens, the cannon fires and thus the railroad car moves in the opposite direction due to conservation of momentum. When the cannonball strikes the opposite end of the car, the movement stops (assuming a perfect situation, of course) since the cannonball is carrying the opposite momentum that the system gained when the cannon was fired. So the car has moved over a slight distance and does so each time a cannon ball is fired. We are temporarily "violating" conservation of momentum over the time that the cannon ball is in the air.

The explosion that gives impulse to the cannon ball is internal to the system, yet causes the overall system to move. The reason for this is that work has been done in firing the cannon ball: chemical energy has produced a mechanical energy. The motion of the system is due to this process.

-Dan

 

[Hurkyl]

We are temporarily "violating" conservation of momentum over the time that the cannon ball is in the air.

The explosion that gives impulse to the cannon ball is internal to the system, yet causes the overall system to move. The reason for this is that work has been done in firing the cannon ball: chemical energy has produced a mechanical energy. The motion of the system is due to this process.

We're not violating conservation of momentum at all.

The net momentum of the system is always zero. The center of mass remains at a fixed location. We moved a mass from one end of the railway car to the other, and therefore the the railway car must move (slightly) in the opposite direction in order to keep the center of mass fixed.

All the chemical energy does is give us the impulse to move the cannonball in the first place.

 

[topsquark]

We're not violating conservation of momentum at all.

The net momentum of the system is always zero. The center of mass remains at a fixed location. We moved a mass from one end of the railway car to the other, and therefore the the railway car must move (slightly) in the opposite direction in order to keep the center of mass fixed.

All the chemical energy does is give us the impulse to move the cannonball in the first place.

That's why I used quotes around "violating." To an untrained eye the system does appear to move! I am aware the CM doesn't move. My point was that there is motion in the system coming from an internal source. I was speculating that that might be something ubavontuba might be considering.

-Dan

 

[Tide]

If you can't (or won't) even try to help me answer my questions, why are you even bothering to post here? Are you purposefully intending to be mean to me?

?

Try this: http://math.ucr.edu/home/baez/noether.html

 

[ubavontuba]

The explosion that gives impulse to the cannon ball is internal to the system, yet causes the overall system to move. The reason for this is that work has been done in firing the cannon ball: chemical energy has produced a mechanical energy. The motion of the system is due to this process.

Well (providing the wheels are frictionless), the boundaries of the car will move but the center of mass remains fixed.

In any event, this is not what I'm interested in. What I'm interested in is understanding the relationships in conservation... vis-a-vis, does a violation of one form (if possible) lead to a possible cascade of violations, or can a violation itself be an isolated incident in relation to the other momentums, forces and energies within the system?

 

[ubavontuba]

We're not violating conservation of momentum at all.

The net momentum of the system is always zero. The center of mass remains at a fixed location. We moved a mass from one end of the railway car to the other, and therefore the the railway car must move (slightly) in the opposite direction in order to keep the center of mass fixed.

All the chemical energy does is give us the impulse to move the cannonball in the first place.

Right, but again off point. I'm not trying to create a reactionless drive or explore possible (not!) variations of a reactionless drive. I'm simply trying to better understand Noether's Theorem. I feel that by examining the repurcussions of a hypothetical violation, I might better understand the relationships of conservation. How the violation is achieved (again, not!) isn't important.

 

[ubavontuba]

Tide,

I've had that link and many others like it in my "Favorites" for a long time. I think I grasp the point of the theorem pretty well. What I don't understand very well is the boundaries between the separately conserved forces, momentums and energies. For instance, were I to use kinetic energy to (illegally) move the railroad car as presented by Hurkyl, must the energy also violate conservation? I don't think so, but I think it can in the case of changing the potential of the isolated system in relation to other systems.

 

[pervect]

There is a derivation of Noether's theorem in Goldstein's "Classical Mechanics".

I'm not sure what your background is, but I think you may be missing the idea that Noether's theorem applies to physical systems which can be derived from an action principle.

This means basically Lagrangian / Hamiltonian mechanics.

Basically, if the Lagrangian is independent of position, momentum automatically pops out as a conserved quantity of motion.

In a nutshell:

Formally, let position be represented by x, and velocity by $\dot{x}$.

Then Lagrange's equation

http://scienceworld.wolfram.com/physics/LagrangesEquations.html

give you the following result directly when $\partial L / \partial x = 0$
(Note that if L is not a function of x, it's derivative with respect to x must be zero!).

$$\frac{d}{dt} ( \frac{\partial L}{\partal \dot{x}} ) = 0$$

This implies that
$$\frac{\partial L}{\partial \dot{x}}$$
which is by defintion momentum, is a conseved quantity (since its time derivative is zero).

This is actually half of the complete theorem, but it should give you an idea of what it's about - given that a system is represneted by a Lagrangian that's independent of position, that system has a conserved quantity, known as the momentum, the partial derivative of the Lagrangian with respect to the velocity. Or in other words space invariance (of the Lagrangian) yields a conserved momentum.

If you don't understand Lagrangian mechanics yet, this is what you need as background for Noether's theorem to make sense.

 

[ubavontuba]

Pervect,

Thanks for your effort. I understand that momentum is conserved though. What I don't understand is; if it wasn't, what would be the consequences?

Let's say (hypothetically speaking) that someone has found a time-dependent momentum wherein the center of mass of an isolated system can be moved off of its center of gravity. Let's further suppose that in this system the aforementioned momentum is clearly time-dependent. That is that like in Hurkyl's example the momentum ceases after internal work ceases, with the exception that the center of mass violates position.

What are the consequences to conservation? Since the isolated system would thusly conserve momentum and energy internally, and externally only attain a change in position, would conservation be technically broken within the system?

Obviously symmetry would be broken externally, as this would change the kinetic energy potentials between the isolated system and other systems, but a reaction wheel can accomplish this in a space vehicle by causing an external arm/appendage to extend in a new direction. Is this not the same thing?

Or, put more bluntly: Is position relevant to conservation in an isolated system?

 

[pervect]

I'm not really sure I am following your hypothetical question correctly. But in order for Noether's theorem to apply, the laws of physics must be given by an action principle.

If you had a physical system that had a momentum that was time-varying, while the system itself was space-symmetric, in violation of Noether's theorem, one could say simply that the laws of physics were not derivable from an action principle.

 

[ubavontuba]

Pervect,

Thanks again for responding.

My hypothetical question is in regards to the interest in asymmetry that so many scientists seem to be pursuing. I'm just trying to apply the principles to a system that's easily understood (for me).

Are you saying that the laws of physics would need to be trashed? I think that might be going too far. Obviously, the laws of physics, as known, work for all known systems (although things like the CP(T) violation, DE, and DM aren't thoroughly understood). Perhaps a slight modification would suffice?

What intrigues me are the common and erstwhile (read professional) attempts to defeat symmetry in various ways. I'm just wondering what the significance would be should someone succeed.

If all the laws of physics are derived from an action principle (I'm assuming you're mostly referring to Newton's laws) and these laws are found to be circumventable, wouldn't that simply open up a new field of study (non-actionable physics?), rather than demolish what has obviously worked?

I have noted a distinct withdrawal from "propellantless propulsion" concepts from the professional scientists, but there still seems to be a lot of interest in it in terms of high-energy physics and multi-dimensional studies. It seems to me that if symmetry is breakable (even if it's only in these high-energy and multi-dimensional applications) that this might eventually point to a method of doing so on a more practical level.

 

[pervect]

Pervect,

If all the laws of physics are derived from an action principle (I'm assuming you're mostly referring to Newton's laws) and these laws are found to be circumventable, wouldn't that simply open up a new field of study (non-actionable physics?), rather than demolish what has obviously worked?

OK, first things first:

I think I may be getting a better idea of what you're asking, and at this point I'm afraid I can't say that the "center of mass" is necessarily a conserved quantity under Noether's theorem. Conservation of momentum, yes - "center of mass", I'd have to think more.

Moving on a bit:

The view of physics based on the action principle underlies all gauge theories, which means basically all of physics as we currently understand it. This has happened because the results from gauge theories has been astonishingly successful in giving correct predictions, and has been easily extended to cover a wide range of physical phenomenon (not only electromagnetism, but the other fundamental forces as well.)

We'd basically have to reformulate just about all of physics in different terms (perhaps closer to traditional F=ma Newtonian physics rather than the current form based on the action principle) to recover. There would be major demolition before rebuilding. A picturesque anology might be a city, leveled by a nuclear blast, being rebuilt. At least that's my opinion.

 

[ubavontuba]

I think I may be getting a better idea of what you're asking, and at this point I'm afraid I can't say that the "center of mass" is necessarily a conserved quantity under Noether's theorem. Conservation of momentum, yes - "center of mass", I'd have to think more.

Please do. I'll be anticipating your thoughts.

Clarification: I'm assuming you mean conservation of the "center of mass" (CM) as in a movement of the CM off of it's original center of gravity (CG). Specifically, an acceleration of the CM away from it's normal, undisturbed equilibrium.

The view of physics based on the action principle underlies all gauge theories, which means basically all of physics as we currently understand it. This has happened because the results from gauge theories has been astonishingly successful in giving correct predictions, and has been easily extended to cover a wide range of physical phenomenon (not only electromagnetism, but the other fundamental forces as well.)

We'd basically have to reformulate just about all of physics in different terms (perhaps closer to traditional F=ma Newtonian physics rather than the current form based on the action principle) to recover. There would be major demolition before rebuilding. A picturesque anology might be a city, leveled by a nuclear blast, being rebuilt. At least that's my opinion.

Clarification: You are referring to "The Least Action Principle" as developed by Maupertuis, Euler, Lagrange, and Hamilton, right? I get it now.

So, are you saying that all of the scientists I've been reading about that are attempting to study/create asymmetry are actually trying to dump the current model? Wo-o-ow... that's heavy.

I wonder what the rebuilding process might reveal? Having to dump the least action principle for all the calculations since Newton certainly would throw a giant monkey-wrench into the works, wouldn't it?

Has anyone attempted to re-examine physics with the assumption that the least action principle isn't correct?

 

[ubavontuba]

By the way, wouldn't the principle of least action describe the path of a curve-ball as a straight trajectory from the pitcher's hand to the catcher's mit?

 

[samalkhaiat]

Has anyone attempted to re-examine physics with the assumption that the least action principle isn't correct?

Yeh, they ended up with garbage such as astrology , but not physics.
The principle of least action is the greatest achievement of theoretical reasoning.It is the most fundamental and the most powerful, principle ever "discovered".

$$\delta{\int_\Omega d^nx \mathcal{L}(x)}=0$$

is an expression of a general behavior in nature. Since the time of DeMaupertuis, practically all observed phenomena have been described by equations shown to be the consequences of a similar principle.
I know of no statement more general, simple and beautiful than DeMaupertuis words (1746);
"Whenever any action occurs in nature, the quantity of action employed by this changed is the least possible"


regards

sam

 

[ubavontuba]

Sam,

Yes, it has been a powerful principle. I just wonder if we might be missing some important details?

As I mentioned above, the least action principle applied to the trajectory of a curve-ball would tend to state a straight trajectory that passes through the batter's bat, rather than curving around the bat, right?

 

[vanesch]

Sam,

Yes, it has been a powerful principle. I just wonder if we might be missing some important details?

As I mentioned above, the least action principle applied to the trajectory of a curve-ball would tend to state a straight trajectory that passes through the batter's bat, rather than curving around the bat, right?

Surely not. If you arrive at that result, it simply means that you haven't the correct action (you're missing certain interaction terms). You need to include all terms that take into account elasticity of the ball, the air flow, and everything that matters to the motion of the ball.

 

[Gokul43201]

As I mentioned above, the least action principle applied to the trajectory of a curve-ball would tend to state a straight trajectory that passes through the batter's bat, rather than curving around the bat, right?

Absolutely not !

Please show us your calculations, and we'll point out the error.

 

[ubavontuba]

Vanesch,

Okay then, how can you be certain you've "accounted for all that matters to the motion of a particle?"

 

[ubavontuba]

Goku143201,

You might as well try and predict the splash patterns of rain droplets!

Here's an interesting http://ame-www.usc.edu/bio/udwadia/papers/Explicit%20equation%20Proc%20of%20RS%20A%202006%20March.pdf I think you might enjoy though.

 

[vanesch]

Vanesch,

Okay then, how can you be certain you've "accounted for all that matters to the motion of a particle?"

When it agrees with the observed motion

 

[samalkhaiat]

Yes, it has been a powerful principle

Still is.

I just wonder if we might be missing some important details?

NO, It is a law of nature.

As I mentioned above, the least action principle applied to the trajectory of a curve-ball would tend to state a straight trajectory that passes through the batter's bat, rather than curving around the bat, right?

WRONG, The actual (observed) trajectory is the one that makes the action stationary.In other words, the actual trajectory is nothing but the solution of the equation of motion which comes out from the action principle.


regards

sam

 

[AlphaNumeric]

By the way, wouldn't the principle of least action describe the path of a curve-ball as a straight trajectory from the pitcher's hand to the catcher's mit?

Lagrangian for the motion of a ball in constant gravity g, vertical direction x, horizontal y would be

$$\mathcal{L} = \frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2}) - mgx$$

Equation of motion for x gives $$\ddot{x} = -g$$

Equation of motion for y gives $$\ddot{y} = 0$$ y is therefore constant.

This neglects air resistence, but the general result is the same, there is a downwards acceleration due to gravity. This is probably the most basic example you could do.

As I mentioned above, the least action principle applied to the trajectory of a curve-ball would tend to state a straight trajectory that passes through the batter's bat, rather than curving around the bat, right?

Nope.

Okay then, how can you be certain you've "accounted for all that matters to the motion of a particle?"

You mean like the force of gravity on a pitchers baseball, like you missed?

 

[ubavontuba]

You mean like the force of gravity on a pitchers baseball, like you missed?

Ouch! You got me! Although technically I didn't state that the ball is thrown on an earthbound baseball diamond, I would certaintly concede that it is implied.

However my point is still valid. The ball goes from the pitcher's hand to the catcher's mit in the course of least action, which might incidently intercept the batter's bat. A seeming paradox.

Of course we know by observation that the ball curved around the bat... but the math didn't tell us.

 

[ZapperZ]

Ouch! You got me! Although technically I didn't state that the ball is thrown on an earthbound baseball diamond, I would certaintly concede that it is implied.

However my point is still valid. The ball goes from the pitcher's hand to the catcher's mit in the course of least action, which might incidently intercept the batter's bat. A seeming paradox.

Of course we know by observation that the ball curved around the bat... but the math didn't tell us.

Unfortunately, you offered ZERO math, not even once, in this thread, to justify your claims. You have not shown any formulation, nor derivation, to back up your claim that there is a "seeming paradox". All you have offered are hand-waving arguments and, worst still, speculation.

Please show a specific derivation using such principle and show that there is a clear problem. Do not do this via innuendos. You have one chance to do this before this thread is locked and you are welcome to continue this in the IR forum.

Zz.

 

[ubavontuba]

Fine.

Equation of motion for y gives $$\ddot{y} = 0$$ y is therefore constant.

Using Lagrangian math, we see by AlphNumeric's example that by his own admission, there is no apparent movement in the y axis. So, does this Lagrangian method give us the full story?

In order to understand the complexities of a curveball pitch, you must use the Magnus effect in fluid dynamics.

Determining a curveball's curve:

FMagnus Force = KwVCv
where:

FMagnus Force is the Magnus Force
K is the Magnus Coefficient
w is the spin frequency measured in rpm
V is the velocity of the ball in mph
Cv is the drag coefficient

P.S. I think I'm all done here. Good-bye.