In my studies to Lagrangian Mechanics I've come across a particular transformation rule: (Wikipedia, Nolting) $$ \delta \mathbf{r}_i = \sum_j \frac{\partial \mathbf{r}_i}{\partial q_j} \delta q_j $$
Which looks a lot like a rule for calculating the differential of a function (from differential geometry): $$ \mathrm{d}f = \sum_j \frac{\partial f}{\partial q_j} \mathrm{d}q_j $$
But as far as I can understand, the first rule is something different. I wanted to derive that, but I'm not sure, whether I made the right conclusions.
In this post an intuition is given for what a virtual displacement is, some extension of the configuration manifold with another parameter $s$ which is independent from the time parameter $t$ (because we define it that way).
Using this definition, I define for each and every point in time $t$ a curve in the configuration space given by $$ \mathbf{r}^\ast_i = \mathbf{r}^\ast_i(q_1(s),q_2(s),\ldots,q_j(s),t) = \mathbf{r}^\ast_i(s) \\ \mathbf{r}^\ast_i(s = 0 ) = \mathbf{r}_i(q_1,q_2,\ldots,q_j,t) $$ with $\mathbf{r}_i$ being the current coordinates along the motion induced by $t$ (saying, $s$ always starts from something that actually happens)
This allows us to define the virtual displacement (again, from the answer in this post) $$ \mathbf{r}_i^\ast(s) - \mathbf{r}_i $$
So far, so normal. Now, the curve defined by $s$ spans a submanifold, and we are interested in its Tangential-Space, meaning we take it's differential restricted to $s$: $$ \mathrm{d}_s(\mathbf{r}_i^\ast(s) - \mathbf{r}_i) = \mathrm{d}_s\mathbf{r}_i^\ast(s) - \mathrm{d}_s\mathbf{r}_i = \mathrm{d}_s\mathbf{r}_i^\ast(s)\\ = \frac{\partial\mathbf{r}_i^\ast(s)}{\partial s} \mathrm{d}s = \frac{\partial\mathbf{r}_i^\ast(q_1(s),q_2(s),\ldots,q_j(s),t)}{\partial s} \mathrm{d}s = \sum_j \frac{\partial\mathbf{r}_i^\ast}{\partial q_j} \frac{\partial q_j}{\partial s}\mathrm{d}s $$ Implying $$ \mathrm{d}\mathbf{r}_i^\ast(s) = \sum_j \frac{\partial\mathbf{r}_i^\ast}{\partial q_j} \frac{\partial q_j}{\partial s}\mathrm{d}s $$ If I now guess correctly, speaking of a differential restricted to a submanifold is a mouthful to say and will be confusing in notation, we intruduce the $\delta$ as a shorthand for this special restricted differential $$ \delta \mathbf{r}_i^\ast(s) = \sum_j \frac{\partial\mathbf{r}_i^\ast}{\partial q_j} \frac{\partial q_j}{\partial s}\delta s =\sum_j \frac{\partial\mathbf{r}_i^\ast}{\partial q_j} \delta q_j \qquad \blacksquare $$
concluding the derivation.
The Question(s):
- Is this derivation correct and did it use the correct terminology?
- Why is there a star ($\mathbf{r}_i^\ast$) in my derivation, but in the original transformation law, there isn't ($\mathbf{r}_i$)? How do we get rid of it?
- My virtual generalized coordinates $\delta q_j$ are tangential components of my curve parameter differential $\mathbf{d}s$, yet in the textbook they are still treated as independent coordinates. Why is that possible?
