The discussion centers on the independence of generalized coordinates and generalized velocities within the framework of Lagrangian mechanics. It emphasizes the analysis of the Lagrangian's functional form to derive the Euler-Lagrange equations, where coordinates and their time derivatives are treated as independent variables. The example provided illustrates a Lagrangian of the form \(L(x, \dot{x}) = \frac{1}{2} m \dot{x}^2 - V(x)\), demonstrating the mathematical legitimacy of treating these variables independently. This foundational understanding is crucial for interpreting phase space diagrams.
PREREQUISITESStudents of physics, mechanical engineers, and researchers in classical mechanics seeking to deepen their understanding of Lagrangian dynamics and its applications in phase space analysis.
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?