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 ith 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.