The discussion revolves around the independence of generalized coordinates and generalized velocities within the context of Lagrangian mechanics and phase space diagrams. Participants explore how these concepts relate to the formulation of the Lagrangian and the derivation of the Euler-Lagrange equations.
The discussion contains multiple viewpoints regarding the independence of generalized coordinates and velocities, with no clear consensus reached among participants.
Participants reference previous discussions and posts, indicating that this topic has been explored multiple times, but specific assumptions or definitions related to independence are not fully articulated.
Well you're apparently a P-F galaxy so can you provide the link to your post regarding this topic?PeroK said:This question has been asked a number of times. Lagrangian mechanics analyses the functional form of the Lagrangian in order to derive the Euler Lagrange equations. At which point the coordinates and derivatives revert to their usual role, with one the time derivative of the other.
If you search my recent posts for the text in italics you should find a more complete answer.
I've never found a way to do it from my phone.VVS2000 said:Well you're apparently a P-F galaxy so can you provide the link to your post regarding this topic?
Ok, if not too much trouble, can you explain how they're independent here itself?PeroK said:I've never found a way to do it from my phone.
You treat them as independent variables and study the functional form of the Lagrangian. In the simplest case you have$$L(x,\dot x) = \frac 1 2 m\dot x^2 - V(x)$$Effectively what you do is say, okay, let's study a function of the form$$L(X,Y) = \frac 1 2 mY^2 - V(X)$$That's perfectly legitimate mathematically.VVS2000 said:Ok, if not too much trouble, can you explain how they're independent here itself?