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Searched this question a lot in different places but could not find any answer that could satisfy me. So I am here to clear my concepts regarding virtual displacement. Now I am clarifying my doubt.

As my textbook "A Student’s Guide to Lagrangians and Hamiltonians" by Hamill says on p.11:

A virtual displacement, $δx_i$, is defined as an infinitesimal, instantaneous displacement of the coordinate $x_i$, consistent with any constraints acting on the system.

And it gives the relationship, $$\delta x_i=\sum_{\alpha=1}^n\frac {\delta x_i}{\delta q_\alpha}$$ The time co-ordinate is frozen in virtual displacement. Now my question is if time is frozen then no displacement should occur. So how the "Virtual displacement" come? What does it mean?

Again $\frac {\delta x_i}{\delta q_\alpha}$ how is it getting the dimension of a displacement? I think the $x$'s and $q$'s dimension is same [L]. So it should be dimensionless.

Please illustrate it so that I do not face any difficulties with virtula-displacement term. I will be too much helpful if you also give an example to show how to calculate virtual displacement.

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