Geometric interpretation of constrained dynamical systems

Question

Below are two pictures from Bojowald's book Canonical Gravity. The author tries to present a geometrical picture of a constrained system, however, the description regarding this seems quite scant to me and I am not able to understand some things regarding this, which I explain below.

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Now, I have the following questions:

1. In the map $(q^i,\dot{q}^i)\mapsto(q^i,p_j(q,\dot{q}))$, what are the $p_j(q,\dot{q})$'s? Are these the momenta corresponding to which the velocities $\dot{q}$'s can be solved for? And how do we construct the coordinates $(q^i,p_j(q,\dot{q}),\psi_{s})$. If $i$ say runs from, say, $1,\dots,n$, and $s$, the number of constraints run from $1,\dots m<n$. Then what values do $j$ take? Let me elaborate with an example. Let the Lagrangian be $$ L = \frac{1}{2}\left(\dot{q}^1 - \dot{q}^2\right)^2 + \frac{1}{2}\left(\dot{q}^3-2\right)^2 + (q^4)^2$$ In this case we have from the definition of momenta $$p_{1} = \dot{q}^1 - \dot{q}^2,~ p_{2} = \dot{q}^2 - \dot{q}^1,~ p_{3} = \dot{q}^3 - 2,~p_{4} = 0 $$ Then the matrix $W_{ij}$ is $$ W_{ij} = \frac{\partial p_{i}}{\partial \dot{q}^j} = \begin{pmatrix} 1 & -1 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} $$ Then $${\rm det}~ W = 0$$ And the rank of the matrix is $2$. Also there are $2$ independent constraints: $$\psi_{1} = p_{1} + p_{2} = 0,~ \psi_{3} = p_{4} = 0.$$ How do I construct the coordinate system $(q^i,p_j(q,\dot{q}),\psi_{s})$? What are $p_{j}$'s?

2. Now, in the caption of the picture there is a remark about fibers away from the costraint surface on which $q^i$ and $p_{j}$ are fixed and some $\dot{q}$ and $\psi_{s}$ vary. Can someone explain this physically and possibly give examples.

I would appreciate it if someone gives an overall explanatuion/picture of what is going on geometrically (that is a map from the original $2n$ dimensional manifold $(q^i,\dot{q}^i)$ to a submanifold (the constraint surface) with lower dimension; the fibres on each point of it etc.) and justify the explanations with examples (not necessarily being limited to the one I chose).

Answer 1

Let us suppress position dependence $q^i$ and explicit time dependence $t$ in the following, and also assume that the Lagrangian $L=L(v)$ is a smooth function of the velocities $v^i$, where $i=1, \ldots, n$. Let us rewrite eq. (3.8) as $$g_i(v)~:=~\frac{\partial L(v)}{\partial v^i}, \qquad i=1, \ldots, n. \tag{3.8}$$ The procedure to perform the possible singular Legendre transformation $v\leadsto p$ is explained in my Phys.SE answer here. The main point is that it is (under certain regularization assumptions) possible to locally divide the $n$ velocity coordinates $$v^i~\longrightarrow~ (u^a,w^{\alpha})$$ and the $n$ momenta coordinates $$p_i~\longrightarrow~ (\pi_a,\rho_{\alpha}) $$ into two types: $r$ of the regular type, and $n-r$ of the singular type. Here $r$ is the rank of the Hessian matrix $$H_{ij}~:=~\frac{\partial^2 L}{\partial v^i \partial v^j}.$$ It is possible to find the inverse velocity-momenta relations $$u^a~=~f^a(\pi,w), \qquad a=1, \ldots, r,$$ forthe regular velocities. Next eliminate the regular velocities $$ h_i(\pi,w) ~:=~ g_i(f(\pi,w),w), \qquad i=1, \ldots, n. $$ One may show that $h_i$ does not actually depend on the $w$-variables. The $n-r$ primary constraints are $$ \phi_{\alpha}(\pi,\rho)~:=~\rho_{\alpha}- h_{\alpha}(\pi)~\approx~0,\quad \alpha=1, \ldots, n-r. $$ Geometrically the primary constraints specify an $(n+r)$-dimensional submanifold in the $2n$-dimensional phase space, which is the cotangent bundle over the $n$-dimensional configuration space.

Now let us return to OP's specific questions.

1. In OP's example it is tempting to use coordinates $(q^+,q^-,q^3,q^4)$ instead where $q^{\pm}=q^1\pm q^2$. Then we have regular velocities $u^-,u^3$ and singular velocities $w^+,w^4$. Similar we have regular momenta $\pi_-,\pi_3$ and singular momenta $\rho_+,\rho_4 $. The 2 primary constraints are $\rho_+\approx 0\approx\rho_4 $.

2. Ref. 1 uses the same notation $p_i$ for the momentum $p_i$, the regular momentum $\pi_a$ and the function $g_i$. Ref. 1 uses the notation $\psi_s$ for the primary constraint functions $\phi_{\alpha}$. The second picture in Fig. 3.2 shows the phase space with coordinates $(q^i,p_j)$. The space in the first picture of Fig. 3.2 with coordinates $(q^i,\dot{q}^j)$ mentioned in the caption is instead the tangent bundle over the configuration space.

   In other words, the caption of Fig. 3.2

   Fig. 3.2 The unconstrained $\color{red}{\text{phase space $P$}}$ with coordinates $(q^i,\dot{q}^i)$ is mapped to the primary constraint surface $r$: $\psi_s=0$ of all points obtained as $(q^i,p_j(q,\dot{q}))$. Dashed lines indicate fibers along which $q^i$ and $p_j$ are fixed but $\color{red}{\text{some $\dot{q}^i$ (and the $\psi_s$)}}$ vary.

   should be changed to

   Fig. 3.2 The unconstrained $\color{green}{\text{tangent bundle in the first picture}}$ with coordinates $(q^i,\dot{q}^i)$ is mapped to the primary constraint surface $r$: $\psi_s=0$ of all points obtained as $(q^i,p_j(q,\dot{q}))$ $\color{green}{\text{in the phase space $P$ of the second picture}}$. Dashed lines indicate fibers along which $q^i$ and $\color{green}{\text{(regular)}}$ $p_j$ are fixed but $\color{green}{\psi_s}$ vary.

References:

1. M. Bojowald, Canonical Gravity and Applications: Cosmology, Black Holes, and Quantum Gravity, 2011.