I've struggled in the past with understanding why we can treat position and velocity as independent variables in the Lagrangian, but I think I may have finally become a bit more enlightened on the subject. I have been reading these notes http://www.physics.usu.edu/torre/6010_Fall_2010/Lectures/01.pdf which to me seem to very well written. However, I would like to check if I'm understanding it all correctly?!
At the bottom of page 5 of the notes the lecturer starts talking about velocity phase space. My question really is, am I understanding this correctly?
Empirically, we know that the state of a physical system at a given instant in time is completely determined if one knows its position and velocity at that instant (where by state it is meant that we know all of the information required to predict the subsequent evolution of the system). The positions that the system can occupy correspond to points on a manifold $\mathcal{M}$. A given point $q\in\mathcal{M}$ can be represented in $\mathbb{R}^{n}$ by a list of $n$-coordinates $\lbrace q_{i}\rbrace$ via a coordinate chart, such that $q= \lbrace q_{i}\rbrace$. Of course, such points exist independently of any curve, which (abusing notation) we denote as $q(t)=\lbrace q_{i}(t)\rbrace$ (where the set of values $\lbrace q_{i}(t)\rbrace$ are the coordinates of the curve in $\mathbb{R}^{n}$ via a coordinate chart). At each point $q\in \mathcal{M}$ there exists a set of tangent vectors $\dot{q}$, with coordinates $\lbrace\dot{q}_{i}\rbrace$ (in some coordinate chart), which form a vector space, tangent to that point, denoted as $T_{q}\mathcal{M}$. Again, these tangent vectors exist independently of any given curve. Furthermore, at each given point on the manifold there exists an infinite number of vectors in the tangent space to the manifold at that point. As such we can introduce a manifold called the velocity phase space in which points on this manifold are determined by the pair $(q, \dot{q})$. Now, for each given point $q$ we can choose from an infinite number of tangent vectors $\dot{q}$ at that point, and as such, the pair are independent variables. Such pairs $(q,\dot{q})$ represent points in configuration space that a curve could pass through, and a possible tangent vector to the curve at that point. Clearly, infinitely many curves in configuration space can correspond to the same $(q,\dot{q})$. Is this ok so far (I'm basically augmenting what is written in the notes with what I think I'm understanding from it)?!
On the middle of page 6 the lecturer pauses to discuss a "common point of confusion" about the notation $\dot{q}$, stating that it is not the derivative of anything, but just a vector that exists at the point $q$. Is this just the case because for each given point we can associate a set of vectors (which form a tangent space at that point); they are not derivatives of anything, but they do, in essence, contain information on what direction (and at what speed) curves can move through that point?!
He then moves on to say that a given curve in velocity phase space looks like $$q=q(t),\qquad\dot{q}=\dot{q}(t)$$ I assume by this that the curve is parameterised by some parameter $t$ and that a given value of $t$ corresponds to a given point $(q,\dot{q})$ in the velocity phase space? Also, he says that such a curve may not actually correspond to any motion of the system and we need $\dot{q}(t)$ to actually represent the tangent to $q(t)$, that is, we need to choose $\dot{q}(t) = \frac{dq(t)}{dt}$. By this does he mean that although a given value of $t$ will correspond to a point $q=q(t)$ and a tangent vector $\dot{q}=\dot{q}(t)$, the tangent vector associated with that value of $t$ may not be that of the curve passing through $q$, hence we require that, in actual fact, the given value of $t$ corresponds to a vector tangent to the point $q$ such that its value $\dot{q}(t)$ corresponds to the derivative of the curve evaluated at that point, i.e. $\dot{q}(t)=\frac{dq(t)}{dt}$?
Having done all this, do we then say that the Lagrangian is a one-parameter family of functions (parameterised by $t$) on the velocity phase space, i.e. $$\mathcal{L}=\mathcal{L}(q,\dot{q}, t)=\mathcal{L}(\lbrace q_{i}\rbrace, \lbrace\dot{q}_{i}\rbrace , t)$$ One may then choose a curve in configuration space $q(t)=\lbrace q_{i}(t) \rbrace$ such that when the Lagrangian is evaluated on that curve, for a given value of $t$ it returns a number $\mathcal{L}(q(t),\dot{q}(t),t)$ corresponding to the value of the Lagrangian evaluated at the point $q=q(t)$ on the curve, whose tangent vector is $\dot{q}(t)=\frac{dq(t)}{dt}$?
Sorry, this probably isn't worded particulary well, but I'm just trying to get my head around the whole idea/ check if I'm understanding it at all correctly?!
