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Should we think of $x(t)$ as a parametric curve and/or a vector valued function?


x(t) is defined to be the position of the point x, so I guess if you want you could define x(t) to be a vector that gives you the displacement from the initial starting point or equivalently as a parametric curve that gives the position given a time. Any function of time that describes the position would satisfy this definition I guess


@dvernet: Yes, $x$ is any differentiable map $x: \mathbb{R} \to \mathbb{R}^n$, i.e., a curve in space, parameterized by time.