Virtual displacement and D'Alembert's principle

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Virtual displacement is a crucial concept in Lagrangian mechanics, representing small spatial changes in a system without any time displacement. It plays a significant role in the application of D'Alembert's principle, which is foundational for deriving the Lagrangian equation. The least action principle states that the actual path taken by a particle is one where the action remains stationary, meaning that virtual displacements do not affect the action. This principle helps in simplifying complex mechanical systems by focusing on energy conservation and motion paths. Understanding virtual displacement enhances the analysis of dynamic systems in classical physics.
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While deriving lagrangian equation using D'Alembert's principle, in Goldstien, they defined a variable called "virtual displacement". Why do we need a concept called virtual displacement? What is its signigicance in classical physics?
 
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Hi,

You can read in wikipedia about it:

http://en.wikipedia.org/wiki/Virtual_displacement

I am not an expert on this matter but from what I have seen, virtual displacements are small spatial displacements r_{i} (with no time displacement) over the path of motion and are used in Lagrangian mechanics.

The least action principle is part of Lagrangian mechanics and states that particles follow a path whose action is stationary, which means that any "small" virtual displacements does not change the action.

You can also read about the least action principle in wikipedia:

http://en.wikipedia.org/wiki/Principle_of_least_action



Sergio
 
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