Goldstein's derivation of E-L equations from D'Alembert

[namphcar22]

In his derivation of the Euler-Lagrange equations from D'Alembert's principle, Goldstein arrives at the expression (equation 1.46) $$\mathbf{v}_i = \frac{d\mathbf{r}_i}{dt} = \sum_k \frac{\partial \mathbf{r}_i}{\partial q_k} \dot{q}_k + \frac{\partial \mathbf{r}_i}{\partial t}$$

where $\mathbf{r}_i = \mathbf{r}_i(q_1, \dots, q_n, t)$ is the position vector of the $i$th particle, as a function of generalized coordinates $q_k$ and time; here the $q_k$'s are also functions of time. We abuse notation since $\mathbf{r}_i$ also represents the embedding of the configuration space of the ith particle in $\mathbb{R}^3$. Later he claims
$\frac{\partial \mathbf{v}_i}{\partial \dot{q}_k} = \frac{\partial \mathbf{r}_i}{\partial q_k}$. Formally this is true, but is this mathematically rigorous? As defined, $\mathbf{v}_i$ is really just a function of time.

 

[Bill_K]

As defined, $\mathbf{v}_i$ is really just a function of time.

No, v is defined as the total time derivative of r, where r is a function of all the q's and t.

 

[namphcar22]

So here's how I'm thinking about it. $\mathbf{r}$ denotes two different things. On one hand, he write $\mathbf{r} = \mathbf{r}(q_1, \dots, q_n, t)$ to denote the embedding of the configuration space in $\mathbb{R}^3$. The $q_i$'s do not depend on time; the $t$-dependence signifies a possibly time-dependent embedding of the configuration space, such as in the case of a bead on a rotating wire. However, when he is thinking of a particular path $\gamma(t)$ of the particle in configuration space, he uses the same symbol $\mathbf{r}$ and also write $\mathbf{r} = \mathbf{r} \circ \gamma$ to denote the embedding of the path in Euclidean space.

By the chain rule, the total time-derivative of $\mathbf{r}$ is $\frac{d\mathbf{r}}{dt} = d\mathbf{r} \circ \frac{d \gamma}{dt}$ where $\mathbf{d\mathbf{r}}$ is the total derivative of $\mathbf{r}$ as an embedding of the configuration space.. Note that $\frac{d\mathbf{r}}{dt}$ is still a function of only time, but $d\mathbf{r}$ is a function on the tangent space.

Goldstein is really using $\mathbf{v}$ to denote both $\frac{d\mathbf{r}}{dt}$ and $d\mathbf{r}$. One one hand, he writes $\mathbf{v} = \frac{d\mathbf{r}}{dt}$. But when he writes $\frac{\partial \mathbf{v}}{\partial \dot{q}}$, he is using $\mathbf{v}$ in the second manner, as a function on the the tangent space.

Is this rationalization correct?

 

[vanhees71]

Yes, that's the usual somwhat sloppy way of physicist's notation.

 

[Bill_K]

Us physicists just write r(q₁(t), ..., q_n(t), t), 'cause we don't know no better.