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) [tex]\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}[/tex]

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

 

[Bill_K]

As defined, [itex]\mathbf{v}_i[/itex] 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. [itex]\mathbf{r}[/itex] denotes two different things. On one hand, he write [itex]\mathbf{r} = \mathbf{r}(q_1, \dots, q_n, t)[/itex] to denote the embedding of the configuration space in [itex]\mathbb{R}^3[/itex]. The [itex]q_i[/itex]'s do not depend on time; the [itex]t[/itex]-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 [itex]\gamma(t)[/itex] of the particle in configuration space, he uses the same symbol [itex]\mathbf{r}[/itex] and also write [itex] \mathbf{r} = \mathbf{r} \circ \gamma[/itex] to denote the embedding of the path in Euclidean space.

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

Goldstein is really using [itex]\mathbf{v}[/itex] to denote both [itex]\frac{d\mathbf{r}}{dt}[/itex] and [itex]d\mathbf{r}[/itex]. One one hand, he writes [itex] \mathbf{v} = \frac{d\mathbf{r}}{dt}[/itex]. But when he writes [itex]\frac{\partial \mathbf{v}}{\partial \dot{q}}[/itex], he is using [itex] \mathbf{v}[/itex] 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.