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?
The concept of independence of generalized coordinates and generalized velocities is a fundamental principle in classical mechanics. It states that the equations of motion of a system can be described using a set of generalized coordinates and their corresponding velocities, which are independent of each other.
Considering independence of generalized coordinates and generalized velocities allows for a more efficient and accurate description of the dynamics of a system. It also simplifies the mathematical calculations involved in solving the equations of motion.
A set of generalized coordinates and velocities are considered independent if they are not related by any constraints or equations. This means that any change in one coordinate or velocity does not affect the others.
Yes, a system can have more generalized coordinates than generalized velocities. This is often the case when there are constraints present in the system, which reduces the number of independent velocities.
The Lagrange equations, which are derived from the principle of least action, take into account the independence of generalized coordinates and velocities. They provide a powerful tool for solving the equations of motion of a system and can be used to determine the dynamics of a system without explicitly considering the forces acting on it.