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General chat for Physics SE (physics.stackexchange.com). For M...
5
Chemistry
Sep 2, 2023 10:46
yep, exactly. it's super hard to read his book, but once you understand what he means, no way you won't be mind blown
Chemistry
Sep 2, 2023 10:44
100%
Chemistry
Sep 2, 2023 10:43
yep
Chemistry
Sep 2, 2023 10:43
he proves why it can't be v^4
Chemistry
Sep 2, 2023 10:42
it's his own logic that you will not meet in any other book
Chemistry
Sep 2, 2023 10:42
yeah, it is. it's mind blowing how he arrives at that
Chemistry
Sep 2, 2023 10:40
but i don't know. maybe in his head, something else was going. he was a genius and who am I to say that his logic is horrible
Chemistry
Sep 2, 2023 10:39
involving newton made things worse
Chemistry
Sep 2, 2023 10:39
well, he did something but then he should have stopped
Chemistry
Sep 2, 2023 10:38
yep. thats my point. if you anyways involve newton, you could say: what my L should be in order to get to newton ?
Chemistry
Sep 2, 2023 10:37
yeah, i know, he just discusses it for free particle (even easier :D)
Chemistry
Sep 2, 2023 10:36
his book is so complex btw, maybe it's me who doesn't get him
Chemistry
Sep 2, 2023 10:35
the long story short, he spends 10 pages of explaining why L = K - U and fails in it because he still uses newton heavily.
Chemistry
Sep 2, 2023 10:34
then he out of nowhere decides that L = K - U (with minus sign)
Chemistry
Sep 2, 2023 10:34
you could have just thought: hm, what shoud my L be, such as EL gives $m\ddot x$
Chemistry
Sep 2, 2023 10:33
he arrives at $L = cv^2$ ($c$ some constant) and he out of nowheere says that $c$ must be $\frac{1}{2}m$, because when you take euler lagrange on it, you should get the same thing as newton does.

this logic is not solid in my opinion, why spending 5 pages on inertial frames, if in the end, you look for function whose erivative matches newton's ($m\ddot x$)
Chemistry
Sep 2, 2023 10:31
you know in the end what he does ?
Chemistry
Sep 2, 2023 10:31
yeah true
Chemistry
Sep 2, 2023 10:31
he uses inertial frame logic
Chemistry
Sep 2, 2023 10:30
he spends 5 pages on explaining why L must be consisting of K
Chemistry
Sep 2, 2023 10:30
not quite
Chemistry
Sep 2, 2023 10:29
but in the end, he just fails in my opinino
Chemistry
Sep 2, 2023 10:29
He has got some interesting proving method about why L = K - U
Chemistry
Sep 2, 2023 10:29
btw, have you read Landau's book ?
Chemistry
Sep 2, 2023 10:28
yeap, it makes sense
Chemistry
Sep 2, 2023 10:27
and if you do that, ofc, you end up with 2 independent variables, but that's not quite the solid proof
Chemistry
Sep 2, 2023 10:26
in your example, you already use the definition of Lagrangian
Chemistry
Sep 2, 2023 10:25
and if $L$ can't be written like that, they $q$ and $\dot q$ definitely are independent
Chemistry
Sep 2, 2023 10:24
yeah, best motivation I found why L can't be written as $q(t), \dot q(t)$ is the partial derivative can't be applied in the end
Chemistry
Sep 2, 2023 10:19
@RyderRude not sure why I can't and why it wouldn't be the same thing
Chemistry
Sep 2, 2023 10:14
@ACuriousMind @RyderRude kudos to you for bearing with me. Thanks so much <3
Chemistry
Sep 2, 2023 10:01
I think I have to stop this and let it go
Chemistry
Sep 2, 2023 10:00
haha, then I don't get what Qmechanics point is about initial condition :D
Chemistry
Sep 2, 2023 10:00
@ACuriousMind then initial condition is not the motivation why $q$ and $\dot q$ are not functions of $t$ in the L.
Chemistry
Sep 2, 2023 09:58
@RyderRude yep, I get that. I'm only arguing how the initial condition is the motivation that $q$ and $\dot q$ are independent
Chemistry
Sep 2, 2023 09:55
it means that in the L, we insert path equation and it's derivative, which means to know L's value at $t=2$, you won't be able to know until you get a path function and insert into it
Chemistry
Sep 2, 2023 09:53
so L could be as well $\frac{1}{2}m\dot q(t)^2 - mgq(t)$
Chemistry
Sep 2, 2023 09:50
"The Lagrangian is supposed to be a function of a state and not just of a path because the energy in a physical system at one point must not depend on its history or future."

and in my case, why would it depend on its history or future ?
Chemistry
Sep 2, 2023 09:48
as if in the L, we directly got path and its derivative's functions
Chemistry
Sep 2, 2023 09:47
imagine $q$ and $\dot q$ in L are functions of $t$
Chemistry
Sep 2, 2023 09:47
ok, then answer this. iF $L = \frac{1}{2}m\dot q(t)^2 - mgq(t)$, can you tell me what this breaks for initial condition motivation ?
Chemistry
Sep 2, 2023 09:45
and at whatever instant, initial condition thing is given, they are independent
Chemistry
Sep 2, 2023 09:45
What Qmechanic says is that position and velocity are independent at initial condition thing which I agree
Chemistry
Sep 2, 2023 09:43
I get why $\dot q$ and $q$ are present in Lagrangian, but yesterday, we were discussing why $q$ and $\dot q$ are not functions of $t$ in the $L$. One of the reasons was if they depend on $t$, then EL won't be able to do partial derivative bcause $L$ is a function of $t$

We all undserstood this reason, but then I saw Qmechanics answer and it seems like he describes second reason/motivation why in L, $q$ and $\dot q$ can't be dependent on $t$. His reasoning is initial condition thing. https://physics.stackexchange.com/a/7802/366606 ("on one hand")
Chemistry
Sep 2, 2023 09:43
exactly, just want to get the initial condition motivation and I'm done
Chemistry
Sep 2, 2023 09:43
@ACuriousMind @RyderRude the reason is since I spent quite amount of time on why theere are independent variables in L, I prefer to fully get it and not just leave it here.
Chemistry
Sep 2, 2023 09:36
can we not discuss 2nd order EL yet ?
Chemistry
Sep 2, 2023 09:36
but in my argument, you can't get L's value until the path is derived.. but why would you need to even know L's value before ?
Chemistry
Sep 2, 2023 09:33
in my argument, initial conditions are still met even if we have $q(t)$ and $\dot q(t)$ - makes sense why ?
Chemistry
Sep 2, 2023 09:32
most of the time, I know equation of motion is 2nd order, so 2 initial conditions needed. $q$ and $\dot q$ at some instant. have you read my argument ? chat.stackexchange.com/transcript/message/64319619#64319619