Interesting! So at least in the case of this specific solution that has time dependence explicitly stated, \frac{\partial}{\partial t}L does disappear! Thanks cyrusabdollahi for pointing that out.
In fact, before taking the partial derivative, if we just expand your expression for L we get...
Thanks r4nd0m for taking the time to offer those clarifications. In any case, as you say, I think I got it straightened out or at least have it at a point where I am comfortable with it. Thanks again for the insight.
Yes, I think I am now beginning to bend my brain around this.
I think my problem was that I was thinking of L as the kind of function that you learn about in beginning-level multivariable calculus, i.e. the kind of environment where you'd encounter the chain rule statement that I mentioned...
At least I hope that's the end of it. Please let me know if it doesn't make sense.
The reason I say this is that there's one nagging thing. If q is really truly independent of t, then why do we have \dot{q}=\frac{d}{dt}q?
An independent variable must have any partial derivative zero...
Okay, thanks. I think this clinched it for me and what you guys are saying now makes sense.
In other words, I'm forgetting that by definition of L, q and \dot{q} are independent variables. As Parlyne explains, this is the whole point of Lagrangian mechanics: looking at different paths by...
Cepheid, I think you might be stabbing in the right direction. In fact I remember now that when the concept of Lagrangians was introduced to me, it was done by treating the displacements as virtual displacements, some kind of hypothetical displacement that happens without any time passing. So...
Hi, thanks for the reply. I think we are talking past each other though. Like you, I'm stating that the total derivative need not be zero. However, I think I see now where my question may have been unclear.
I'll try rephrasing, and will state the question clearly at the end.
We start with...
I'm trying to understand something that's coming from my Marion & Thornton (4th edition 1995 on p. 264 in a section titled "Conservation Theorems Revisited"). The topic is conservation of energy and introduction of the Hamiltonian from Lagrange's equations.
We're told that the Lagrangian...