Sujet : Re: The momentum - a cotangent vector?
De : ram (at) *nospam* zedat.fu-berlin.de (Stefan Ram)
Groupes : sci.physics.researchDate : 10. Aug 2024, 07:16:00
Autres entêtes
Organisation : Stefan Ram
Message-ID : <energy-20240809183514@ram.dialup.fu-berlin.de>
References : 1
ram@zedat.fu-berlin.de (Stefan Ram) schrieb oder zitierte:
How can I see that (given that q' is a tangential vector)
p is a cotangential vector?
Here's a little calculation I whipped up in the realm of good
old classical mechanics, no relativity involved.
I'm starting with the well-known formula
E = 1/2 m v^2
Using Cartan's calculus, from this, I come up with:
dE = m v dv + 1/2 v^2 dm.
And since dm = 0 (I assume the mass is constant):
dE = p dv.
Now let's write out the implied scalar product as "*":
dE = p * dv.
This "p *" is now a covector acting like a linear function, mapping
changes in velocity (a vector) to changes in energy (a scalar).
BTW, we also can derive the "other" relationship dE = v dp!
Writing "1/2 m v^2" as "1/2 m v v", we can see that
E = 1/2 p v
, so,
dE = 1/2 p dv + 1/2 v dp
. But since we already had established that dE is "p dv" for a
constant mass m, "1/2 p dv" must be "1/2 dE", so that,
dE = 1/2 dE + 1/2 v dp.
Subtracting "1/2 dE" on both sides gives:
1/2 dE = 1/2 v dp,
and multiplication by 2,
dE = v dp.
So, dE is both "v dp" and "p dv" when the mass m is constant!