# Position in generalized coordinates

In Lagrangian mechanics, when talking about a particle position expressed in generalized coordinates it is usual to find the expression:

$$\mathbf{r}(q_0,...,q_k,t)\tag{1}$$

what it means this isolated $$t$$ time variable?

Wikipedia uses the expression (see here):

$$\mathbf{r}(\mathbf{q}(t))\tag{2}$$

I can understand all these expressions as equivalent:

$$\mathbf{r}(t) = \mathbf{r}(\mathbf{q}(t)) = \mathbf{r}((q_0,...,q_k)(t)) = \mathbf{r}(q_0(t),...,q_k(t)) = \mathbf{r}(q_0,...,q_k)$$

being the last one a typographic simplification. But not equivalent to $$\mathbf{r}(q_0,...,q_k,t)$$

Note we are not talking about a time-dependent vector field defined in a space like in $$\mathbf{r}(x,y,z,t)$$, but about the coordinates of particles.

Usual expressions from $$\mathbf{r}(q_0,...,q_k,t)$$ use chain rule including $$t$$ as an independent variable, like in: $$d\mathbf{r} = \sum_k \frac{\partial \mathbf{r}}{\partial q_k} dq_k + \frac{\partial \mathbf{r}}{\partial t} dt$$.

• You can consider things like $\vec{r}' (t) = \vec{r} + \vec{v}\,t$ as pretty normal transformations... I guess the notation just points to a moving frame while changing generalized position transformations as well. Jul 2, 2020 at 19:39
• I think the basic question behind your doubt is explained well enough in this PSE answer at: physics.stackexchange.com/q/9122. In other words, the difference is in a choice of \textit{representation} where the time dependence is explicit or implicit. At a mathematical level it will introduce extra terms in the Lagrangian and action variation. Jul 2, 2020 at 20:08
• @Lelouch: sorry, I do not understand your comment. Obviously, it is not the same $f(x,y)$ than $f(x)$ nor $f(x(y))$. About the link, I do not see direct relation with this question, the linked question seems to ask about difference between derivative and partial derivative. Jul 3, 2020 at 18:13

$${\bf r}_i(q^1,\ldots,q^n,t)~\in~ \mathbb{R}^3$$
of the $$i$$'th point particle, $$i\in\{1,\ldots,N\}$$, can depend on $$n$$ generalized coordinates $$q^1,\ldots,q^n,$$ and explicitly (as well as implicitly) on time $$t$$. In other words, we assume $$3N-n$$ holonomic constraints.