Virtual Displacement: Analyzing Langrange Equations

[wormhole]

hey, all

i'm now studiynd analytical mechanics and the subject is Langrange equations. What i can't grasp is the meaning of virtual displacement term.

The formal definition says that:
it's a small displacement of particle with agreement to constraints in such a way that no time passes and uknown forces don't change. The displacement itself has no relation to actual particle path.

there are two places where i get confused:

1) the definition itself when it says that "no time passes"

2) the actual calculation of virtual displacement where i don't understand the reason why the differentiation is done only with respect to generalized coordinates (q) and time is ignored...

Xj - cartezian coordinates
delta(Xj) - virtual displacement of Xj
Qk - generalized coordinates

 

[turin]

Look at your identical post in www.advancedphysics.org[/URL]

 

[GSXR750]

Hi there,

Virtual displacement is a concept used in analytical mechanics to analyze the motion of a system. It refers to a hypothetical displacement of a particle that satisfies the constraints of the system and does not involve any actual physical movement of the particle. This means that the particle does not actually move from its initial position, but we can analyze its behavior as if it did.

In the definition you mentioned, "no time passes" simply means that the virtual displacement is instantaneous and does not involve any change in time. It is a small, instantaneous displacement that allows us to analyze the system at a particular moment in time.

As for your second confusion, the calculation of virtual displacement is done with respect to generalized coordinates because these are the variables that describe the configuration of the system. Time is not considered because we are only interested in the instantaneous behavior of the system, and time is not a variable that describes the configuration of the system. Therefore, we only differentiate with respect to the generalized coordinates to analyze the virtual displacement.

I hope this helps to clarify the concept of virtual displacement for you. Keep studying and practicing, and it will become clearer as you continue to learn about Langrange equations. Good luck!