Virtual Displacement: Definition & Meaning

In summary, the concept of virtual displacement is introduced in Goldstein's book on Classical Mechanics, where it is defined as a very small displacement that is not visible. It is different from ordinary displacement in that it occurs in an extra time dimension, called s-time, which is independent of the normal t-time. The purpose of considering virtual displacement is to generate possible paths for the system, with each path corresponding to a fixed value of s. This concept is further explored in Chapter 2 of the book and is related to Lagrange's equations.
  • #1
niyazkc88
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I have been reading Classical Mechanics by goldstein for over 2 months. I have encountered the concept of Virtual Disolacement in that book. Some of my friends have told me that it means a very small desplacement that you cannot see..
However, I am not sure if it is right. Then what is the difference between virtual and ordinary displacement? Can anyone please tell me what virtual displacement really means??
 
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  • #2
I understand you. I also read Goldstein now and then, and I asked myself the same question. But I think it gets a little more clear in Chapter 2, about the Hamiltonian formulation.

Imagine that there is an extra time dimension, call it s, that is independent of the ordinary time t, and imagine that the system can move in s-time as well as in t-time. The constraint equations (holonomic) may contain t explicitly, but not s.

So, when the system moves in s-time, for a fixed t, the contraints will only depend explicitly upon the space coordinates. We obtain the actual motion in time t for the system if we keep s=0.

A virtual dispacement [itex]\delta q[/itex] of some generarlized space coordinate q can then be interpreted as a differential displacement of q in s-time: [itex]\delta q = \frac{\partial q}{\partial s}ds[/itex], and actually, in every formula where [itex]\delta q[/itex] occurs, we can use [itex]\frac{\partial q}{\partial s}[/itex], evaluated at s=0, instead. The formulas we obtain in this way shall be valid for all possible motions of the system in s-time, and therfore, for all possible values of the [itex]\frac{\partial q}{\partial s}[/itex], for independent q:s. This will lead to Lagrange's equations just as in Chapter 1 in Goldstein.
 
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  • #3
I have a few questions regarding your explanations, Erland.
How can you bring 2 time dimensions independent of each other? What does it mean? I mean time is only 1 dimension..
So youre saying that virtual displacement is the displacement in s-time, which is different from the normal t- time.. But that's still bothering me..Because I can't properly understand what s-time is.
Can you tell me what the differnce is between t time and s time??
 
  • #4
Well, the s-time doesn't exist in the real world, it's just imagination, that why it's called virtual displacement. And it is not necessary to consider it as a time-coordinate (that's just how I personally think of it), but as any parameter which the generalized coordinates is imagined to depend upon, so that instead of q(t) we have q(t,s). In Chapter 2 in Goldstein, this parameter is called [itex]\alpha[/itex]. We consider the system as moving wrt both ordinary time and s-time, and consider displacements wrt s (it is probably because when I think of motion, I think of it wrt time, that I consider s as a kind of time coordinate). But it doesn't exist in reality, it is just a way to generate possible paths, one path for each fixed value of s, and the actual path will be that one for which the derivative wrt s of the integral of the lagrangian vanishes. Look up Chapters 1 and 2 in Goldstein.
 
  • #5


Virtual displacement is a term used in classical mechanics to describe a very small, hypothetical displacement that is considered in order to analyze the behavior of a system. It is not an actual physical displacement that can be observed, but rather a theoretical concept used in calculations and equations.

The main difference between virtual and ordinary displacement is that virtual displacement is a purely mathematical concept, while ordinary displacement refers to a physical change in position. Virtual displacement is often used in the study of systems in equilibrium, where small changes in position can help determine the stability and behavior of the system.

In simpler terms, virtual displacement allows us to understand how a system would behave if it were slightly disturbed, without actually physically disturbing it. It is a useful tool in theoretical analysis and can help us make predictions about the behavior of physical systems.

I would also like to add that the concept of virtual displacement is not unique to classical mechanics, but is also used in other fields such as thermodynamics and quantum mechanics. It is an important concept to understand for anyone studying physics or engineering. I hope this helps clarify the meaning of virtual displacement.
 

1. What is Virtual Displacement?

Virtual displacement is a concept in physics that refers to a hypothetical infinitesimal change in the position of an object or system. It is used to analyze the behavior of systems in equilibrium and predict their stability.

2. How is Virtual Displacement different from actual displacement?

The main difference between virtual displacement and actual displacement is that virtual displacement is a hypothetical concept used for analysis, while actual displacement is a physical change in the position of an object or system. Virtual displacement does not involve any actual movement, but rather a theoretical change in position.

3. What is the significance of Virtual Displacement in physics?

Virtual displacement is important in physics because it allows us to analyze and predict the behavior of systems in equilibrium. It helps us understand the stability of objects and systems, and how they respond to external forces.

4. Can Virtual Displacement be observed or measured?

No, virtual displacement cannot be observed or measured because it is a theoretical concept. It is used in mathematical calculations and models to understand the behavior of systems, but it does not involve any actual physical change.

5. How is Virtual Displacement used in real-world applications?

Virtual displacement is used in various fields such as engineering, mechanics, and physics to analyze the stability and behavior of systems. It is also used in computer simulations to model and predict the motion of objects and systems. Additionally, it is used in the design and optimization of structures and machines to ensure their stability and safety.

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