I am reading lecture notes on mathematical physics and I have a question regarding the following passage:
In classical mechanics, the state of a particle at time $t$ is uniquely determined by its position $x(t)$ and its velocity $\dot{x}(t)$, or equivalently its momentum $\xi(t)$. Mathematically the difference between the two points of view if the ambient space is $\mathbb{R}^n$, then $\dot{x}(t)\in T\mathbb{R}^n$ is a tangent vector while $\xi(t)\in T^*\mathbb{R}^n$ is a cotangent vector. The phase space of classical mechanics is the cotangent space $T^*\mathbb{R}^n\cong \mathbb{R}_{\xi}^n\times\mathbb{R}_x^n$.
So if I understood everything correctly, we have two options: Either 1.) the state of a particle at time $t$ is given by $(x(t), \dot{x}(t))\in\mathbb{R}^{2n}$ or $(x(t), \xi(t))\in\mathbb{R}^n\times T^*\mathbb{R}^n$, since $\xi$ is a cotangent vector. But then, if $T^*\mathbb{R}^n\cong \mathbb{R}^{2n}$, isn't one possible phase space $\mathbb{R}^n\times T^*\mathbb{R}^n\cong \mathbb{R}^{3n}$? Or is there some minor error in the passage or have I just not understood something?