Understanding Noether's Theorem: Conservation of Momentum and Energy

[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 resistance, 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.

 

[AlphaNumeric]

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

Obviously if there were no external forces applied to it, then the ball would travel in a straight line, but when who plays blaseball in zero gravity. All you've got to do it hit is upwards (from your perspective) and it'll never come down, a certain home run.

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.

What?! I don't see what you're getting at. Are you saying that if you put a batter between the pitcher and the catcher who attempts to hit the ball the principle of least action says no matter what he does, the batter will miss? Are you deliberately attempting to just alter the situation and claim something completely stupid?

If you model the balls motion under gravity, it follows a parabola, my last post shows that, saying ' but the math didn't tell us' is just being ignorant of what I and many others have posted. It does not follow a straight line!

The maths won't tell us for certain if the batter will hit the ball, otherwise you'd not get people betting on baseball. Once he has hit it, the maths will model the trajectory from the bat to wherever it lands.

You've been asking all these questions in an attempt to understand Noether's Theorem more, but have you actually done any of the maths? Derived any equations of motion for things in gravity, electric fields, heck, even quantum fields (ie the Dirac equation)? I'd wager not, or you'd not be taking what you are about such an elementary system as a ball in gravity.

 

[ZapperZ]

Fine.



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.

But this is awfully silly. It is the shorcoming of YOUR model, not Lagrangian mechanics! You could have easily done this in Newtonian force equation and forgetting to put ALL the necessary forces. You'll end up with the SAME shortcomming.

It is why when anyone writes a paper to describe a phenomena, the Hamiltonian or Lagrangian is ALWAYS shown. This will clearly indicate to everyone reading it what interactions are being considered and what are being left out. You NEVER wrote yours. If you had done so and left out this interaction, and THEN complain that the method failed, everyone would have said "Well DUH, you didn't include such-and-such in your Lagrangian". This thread would have ended in just one page.

Instead, you left it to others to try and decipher what exactly are the scope of the interactions you wish to consider in such a situation. It is my conclusion that you didn't wish to ask anything or learn anything with this thread.

As promised, this thread is DONE!