Jump to content

Noether's Theorem and dimensional analysis

Recommended Posts

Hi,

How can Noether's Theorem be reconciled with dimensional analysis?

For example, invariance under spatial translation is equivalent to
conservation of momentum.  Spacial translation has no involvement of
mass but mass is essential to momentum.

How can the invariance independent of mass be equivalent to
conservation of a quantity where mass is essential?

Thanks,                 ... Peter E.

Share on other sites

5 hours ago, Peter Easthope said:

Hi,

How can Noether's Theorem be reconciled with dimensional analysis?

Why does it need to be?

5 hours ago, Peter Easthope said:

For example, invariance under spatial translation is equivalent to
conservation of momentum.  Spacial translation has no involvement of
mass but mass is essential to momentum.

Photons disagree with you.

5 hours ago, Peter Easthope said:

How can the invariance independent of mass be equivalent to
conservation of a quantity where mass is essential?

Thanks,                 ... Peter E.

Neither mass nor velocity would change under a spatial translation

Share on other sites

6 hours ago, Peter Easthope said:

Hi,

How can Noether's Theorem be reconciled with dimensional analysis?

For example, invariance under spatial translation is equivalent to
conservation of momentum.  Spacial translation has no involvement of
mass but mass is essential to momentum.

How can the invariance independent of mass be equivalent to
conservation of a quantity where mass is essential?

Thanks,                 ... Peter E.

28 minutes ago, swansont said:

Why does it need to be?

Indeed.

Perhaps you mean symmetries, relate to conservation laws.

In your particular case are you referring to this ?

Quote

Herman Weyl

The theorem of conservation of energy is only one component, the time component, of a law which is invariant for Lorenz trnasformations, the other components being the space components which express the conservation of momentum.

The total energy as well as the total momentum remains unchanged; ......

The symmetry of the four dimensional energy momentum tensor tells us the the momentum density = 1/c2 times the energy flux.

But your question may have nothing to do with the symmetries (and invariances) of this particular transformation.

So please provide the context in which you have asked this question.

Edited by studiot
Share on other sites

Quote

How can Noether's Theorem be reconciled with dimensional analysis?

Quote

Why does it need to be?

Only an interest.  Not a necessity.

Quote

Photons disagree with you.

Good point.  Several details to understand.  More below.

Quote

The theorem of conservation of energy is only one component, the time

component, of a law which is invariant for Lorenz trnasformations, the
other components being the space components which express the
conservation of momentum.

The total energy as well as the total momentum remains unchanged;
......

The symmetry of the four dimensional energy momentum tensor tells us
the the momentum density = 1/c2 times the energy flux.

Thanks.  Definitely helpful.

Quote

But your question may have nothing to do with the symmetries (and
invariances) of this particular transformation.

Yes, my understanding is meagre.

Quote

So please provide the context in which you have asked this question.

I don't have a specific context.  Just trying to understand the theorem.

The conservations, C, involve spacetime and mass whereas the
symmetries, S, involve spacetime but not mass.

Equivalence requires two implications: C => S and S => C.

C => S entails removal of mass.
S => C entails introduction of mass.

The presentations I've seen avoid attention to mass. I'm not
claiming that's incorrect; only that I'm interested to understand
the involvement of mass better.

Thanks for the feedback,                    ... P.

Share on other sites

37 minutes ago, Peter Easthope said:

The conservations, C, involve spacetime and mass whereas the
symmetries, S, involve spacetime but not mass.

The conserved quantities are certain properties of particles or systems - such as linear and angular momentum, and energy. Not spacetime. (also, mass is not a conserved quantity. This is the invariant mass, not the so-called relativistic mass, which is a proxy for energy)

The symmetries involve whatever spacetime symmetry you are looking at  - translation in space (momentum), translation in time (energy) and rotation (angular momentum).

Share on other sites

2 hours ago, Peter Easthope said:

I don't have a specific context.  Just trying to understand the theorem.

So tell us what expression of the theorem you have as it is difficult to offer generalisations if the question is too broad.

Swansont has offered a good summary in his second post +1

Share on other sites

20 hours ago, studiot said:

So tell us what expression of the theorem you have as it is difficult to offer generalisations if the question is too broad.

A convenient elementary example: a Newtonian CO2 puck on a level table.  Non-zero invariant mass, m.

(S) Verification of time invariance: verify that the path is a straight line; verify that velocity is constant.  Assume m is constant or ignore it?

(C) Conservation of energy: assuming the table is level, gravitational potential is constant.  Evaluate (m v^2)/2 over a time sequence and verify constancy.  Assume m is constant?  Otherwise, how is m measured during motion?

The invariant mass is assumed to be constant in S?  Also in C?

Thx,          ... P.

Make that "air puck" rather than "CO2 puck",       ... P.

Share on other sites

1 hour ago, Peter Easthope said:

A convenient elementary example: a Newtonian CO2 puck on a level table.  Non-zero invariant mass, m.

(S) Verification of time invariance: verify that the path is a straight line; verify that velocity is constant.  Assume m is constant or ignore it?

(C) Conservation of energy: assuming the table is level, gravitational potential is constant.  Evaluate (m v^2)/2 over a time sequence and verify constancy.  Assume m is constant?  Otherwise, how is m measured during motion?

The invariant mass is assumed to be constant in S?  Also in C?

Thx,          ... P.

Make that "air puck" rather than "CO2 puck",       ... P.

Is the mass varying with position in the first example? If it is, then momentum is not conserved for the object, and we can confirm this with Newton’s laws of motion. There must be an external force on it.

Is the mass varying with time in the second example? If it is, mechanics or relativity tells us that its energy isn’t conserved. There must be some interaction (e.g. work is done) to account for this.

You wouldn’t be applying symmetry to these problems if mass were varying, as their Lagrangians are not constant, i.e. not symmetric under the relevant translation.

Share on other sites

1 hour ago, swansont said:

Is the mass varying with position in the first example?

Is the mass varying with time in the second example?

I imagined constancy of mass or masses

1 hour ago, swansont said:

You wouldn’t be applying symmetry to these problems if mass were varying, as their Lagrangians are not constant, i.e. not symmetric under the relevant translation.

OK, thanks.  I need to understand the Lagrangian calculation better,       ... P.

Share on other sites

I need to study the Lagrangian calculation more.

Unfortunately the engineering curriculum focuses on "practical" calculations at the expense of the abstract formulations.

Thanks for the help,         ... P.

Share on other sites

4 hours ago, Peter Easthope said:

I imagined constancy of mass or masses

OK, thanks.  I need to understand the Lagrangian calculation better,       ... P.

Again swansont has hit the nail exactly on the head here.

A carbon dioxide puck would be continually subliming away like crazy.
So its mass would be continually diminishing.

From the text of your original submission I wonder if you realise that 'space' alone does not admit of translation.
Something translates or is translated in or through space (or rotates in it).
And it is the invariance of some property of that something that provides the symmetry, not the space itself.

Did you get your reference to Noether from somewhere like this ?

Edited by studiot
Share on other sites

Quote

... 'space' alone does not admit of translation.
Something translates or is translated in or through space (or rotates in it).

An instance or example: do an experiment in London, Ontario and the same experiment in London, England.  Assuming conditions are genuinely duplicated (or that bias is avoided) the results in the two cases will be the same.

8 hours ago, studiot said:

Did you get your reference to Noether from somewhere like this ?

Didn't find the source for that but it might be under copyright.  This is readily available.  https://en.wikipedia.org/wiki/Noether's_theorem

Regards,      ... P.

Share on other sites

2 hours ago, Peter Easthope said:

An instance or example: do an experiment in London, Ontario and the same experiment in London, England.  Assuming conditions are genuinely duplicated (or that bias is avoided) the results in the two cases will be the same.

A fine example of my point.

"an experiment"  is something other than space.

A simpler example would be congruent triangles in euclidian geometry.

Congruence is a symmetry that can be established by the conventional methods as taught in lower high school.

But the whole of euclidian geometry can be recast in terms of transformations so that congruence becomes a translation (and perhaps a rotation) so that you can overlay one triangle on another.

But you require something other than space alone, in this case two triangles, as I said.

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now
×
×
• Create New...

Important Information

We have placed cookies on your device to help make this website better. You can adjust your cookie settings, otherwise we'll assume you're okay to continue.