Virtual Displacement: Analyzing Langrange Equations

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Virtual displacement refers to a hypothetical small change in a particle's position that adheres to constraints without the passage of time or changes in unknown forces. The confusion arises from the phrase "no time passes," which emphasizes that these displacements are instantaneous and not tied to the actual motion of the particle. Additionally, when calculating virtual displacement, differentiation is performed solely with respect to generalized coordinates, ignoring time, to simplify the analysis of systems under constraints. Understanding these concepts is crucial for applying Lagrange equations effectively in analytical mechanics. Clarifying these points can enhance comprehension of virtual displacements in the context of constrained motion.
wormhole
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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

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