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Q&A

Why does this definition of generalized forces work?

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I don't understand an equation in the discussion of Lagrangian Mechanics in the book Mathematical Aspects of Classical and Celestial Mechanics by Arnold et al. (section 1.2.1).

For context, they are considering a set of $n$ particles. The set of functions $\mathbf{r}_i:\mathbb{R}\to\mathbb{R}^3$ give the position of each particle as a function of time. The particles are constrained to a manifold $M$ embedded in $\mathbb{R}^{3n}$ so that $\mathbf{r}=(\mathbf{r}_1(t),...,\mathbf{r}_n(t))\in M$ for all $t$.

They say "let $q=(q_1,...,q_k)$ be local coordinates on $M$." I'm not sure what exactly they mean by this. They treat $q$ as a function of $t$, so maybe they mean $q=\varphi\circ\mathbf{r}$, where $\varphi:U\subset M\to \mathbb{R}^k$ is a chart that covers the area of interest.

The Question: Given vector $\mathbf{F}=(\mathbf{F_1},...,\mathbf{F_n})\in\mathbb{R}^{3n}$, they define the generalized force covectors $Q(q)$ by the equality $$\sum_{i=1}^n\langle F_i, d\mathbf{r_i}\rangle=\sum_{j=1}^k Q_j dq_j.$$ Is this well-defined? How can we define a covector just by its inner product with a single vector?

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This expression is not the clearest way of writing this, but the idea is that we are defining the components of the covector $Q(q)$ on a basis of differential 1-forms $dq_j$, i.e. $Q = \sum_{j=1}^k Q_j dq_j$. This is made more clear by Wikipedia's expression of this statement: $Q_j = \sum_{i=1}^n \langle \mathbf F_i, \frac{\partial \mathbf r_i}{\partial q_j}\rangle$. We get the result from the book via: $$\begin{align} \sum_{j=1}^k Q_j dq_j & = \sum_{j=1}^k \sum_{i=1}^n \left\langle \mathbf F_i, \frac{\partial \mathbf r_i}{\partial q_j}\right\rangle dq_j \\ & = \sum_{i=1}^n \sum_{j=1}^k \left\langle \mathbf F_i, \frac{\partial \mathbf r_i}{\partial q_j}\right\rangle dq_j \\ & = \sum_{i=1}^n \left\langle \mathbf F_i, \sum_{j=1}^k \frac{\partial \mathbf r_i}{\partial q_j} dq_j\right\rangle \\ & = \sum_{i=1}^n \left\langle \mathbf F_i, d\mathbf r_i\right\rangle \end{align}$$ where the final equality uses the differential calculus expression of a differential 1-form in terms of basis 1-forms: $d\mathbf f = \sum_{j=1}^k \frac{\partial \mathbf f}{\partial x_j}dx_j$. It's a little confusing but not incorrect to explicitly write out the components on one side but incorporate them on the other. This is compounded by the fact that the differential 1-form $d\mathbf r_i$, among other relevant concepts, is never explicitly defined in the book (at least prior to here). It doesn't need to give a ground up reconstruction of differential manifolds, but it probably wouldn't have hurt for it to have spent an introductory section or at least an appendix setting notation and terminology for it.

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Oh, I see. I was mistakenly thinking of $Q_j$ as a vector instead of a number. JoshuaTS‭ 5 months ago

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