Unlocking Nature's Secrets Through Symmetry and Transformation

[actionintegral]

These forums help me practice my communication skills, thank you to those who helped me reword the following little document.
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I am going to try and deduce as much as I can about the nature of matter from symmetry and the galilean

transformation alone.

Suppose two identical objects approach each other with equal and opposite velocity v.

They meet in front of me.

What will happen next?

Whatever happens, we would expect the same thing to happen to both objects, since they are identical.

Let's imagine that the two objects stop dead.

Now witness this same event from the point of view of an observer moving along with one of the objects.

This observer will see one object at rest, and the other object approaching at -2v.

After the two objects meet, this observer sees them both move away together at -v.

This might be called "conservation of momentum".

Now, I argue that this phenomenon, where the two combined objects appear to move away more slowly is a result

of the galilean transformation and the change in velocity alone. To see this, replace the moving objects with
pixels on a computer screen, dots of light on the wall, or any objects that move toward each other. The only

important thing is that they stop dead upon meeting. The reason for their stopping is immaterial.

Now witness this same event from the point of view of an observer moving along with one of the objects. This

observer will see one object at rest, and the other object approach at -2v.

After the two objects meet, this observer sees them move away together at -v!

You and I know the secret, that the objects really don't have any momentum, and their "collision" is

contrived.

But this observer, not knowing differently, might call that phenomenon "conservation of momentum".

He would claim that the combined object has a mass of 2m, and therefore is moving away at -v in order that
momentum might be conserved.

It is possible to conclude that "conservation of momentum" is really an artifact of the changes in velocity

and the galilean transformation. Of course can be extended to the Lorentz transformation.

 

[pmb_phy]

Speaking of conservation of momentum, you might find this interesting. Given the law of conservation of 3-momentum one can derive the conservation of mass (energy) relation (making it a theorem rather than a law. For details see

http://www.geocities.com/physics_world/sr/conservation_of_mass.htm

Pete

 

[nrqed]

These forums help me practice my communication skills, thank you to those who helped me reword the following little document.
__________________________________________________________

I am going to try and deduce as much as I can about the nature of matter from symmetry and the galilean

transformation alone.

Suppose two identical objects approach each other with equal and opposite velocity v.

They meet in front of me.

What will happen next?

Whatever happens, we would expect the same thing to happen to both objects, since they are identical.

Let's imagine that the two objects stop dead.

Now witness this same event from the point of view of an observer moving along with one of the objects.

This observer will see one object at rest, and the other object approaching at -2v.

After the two objects meet, this observer sees them both move away together at -v.

This might be called "conservation of momentum".

Now, I argue that this phenomenon, where the two combined objects appear to move away more slowly is a result

of the galilean transformation and the change in velocity alone. To see this, replace the moving objects with
pixels on a computer screen, dots of light on the wall, or any objects that move toward each other. The only

important thing is that they stop dead upon meeting. The reason for their stopping is immaterial.

Now witness this same event from the point of view of an observer moving along with one of the objects. This

observer will see one object at rest, and the other object approach at -2v.

After the two objects meet, this observer sees them move away together at -v!

You and I know the secret, that the objects really don't have any momentum, and their "collision" is

contrived.

But this observer, not knowing differently, might call that phenomenon "conservation of momentum".

He would claim that the combined object has a mass of 2m, and therefore is moving away at -v in order that
momentum might be conserved.

It is possible to conclude that "conservation of momentum" is really an artifact of the changes in velocity

and the galilean transformation. Of course can be extended to the Lorentz transformation.

The problem I have with this is that it is a far too restrictive example. Your discussion can not be used to conclude that momentum is conserved in the way it is usually formulated (for arbitrary masses).
Try to prove that $m_1 {\vec v_1} + m_2 {\vec v_2}$ is conserved for arbitrary masses this way! The only way to do it is to start from Newton's laws and work within a certain approximation.

 

[actionintegral]

http://www.geocities.com/physics_world/sr/conservation_of_mass.htm
Pete

Thanks for sharing that link, Pete. Actually, it was a book on special relativity that gave me the idea. It was talking about how "relativistic mass" can be derived from symmetry and the transformation of velocities alone. "Relativistic mass" arises as a useful fiction.

 

[Quaoar]

Hate to be nit-picky but the paragraph breaks make the original post very difficult to read... :)

 

[actionintegral]

Hate to be nit-picky but the paragraph breaks make the original post very difficult to read... :)

I appreciate the feedback. I did a lazy cut and paste from notepad.

 

[actionintegral]

Your discussion can not be used to conclude that momentum is conserved ... (for arbitrary masses).
QUOTE]

Thank you for your thoughtful response. You are correct, of course. I think of this as the "locomotive vs. mosquito" problem. But I would like to share with you my tiny steps in this direction, with your indulgence:

First, I need to prove a trivial lemma - Imagine I am at rest and I watch two identical objects with equal and opposite velocities "v" meet and stop dead. Now imagine another observer witness the same event while moving past me at 3v.

He will report two objects moving past him, one object overtaking the other at twice the velocity. The conglomeration of the two will move away more slowly in such a way that "momentum" is conserved.

As before, the "mass" here can be fictitious; the "collision" can be faked.