# Geometric interpretation of constrained dynamical systems

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.  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 $$1,\dots m. Then what values do $$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).

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.

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.
• "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." Can you please elaborate on this point? That is how we can understand this given the definitions of tangent bundle etc., and why along the fibers velocities $\dot{q}$ and $\psi_{s}$ vary but not coordinates and momenta, etc. as the book mentions. Jun 18 at 1:59
• I updated the answer. Jun 18 at 8:56
• "Ref. 1 uses the same notation $p_i$ for the momentum $p_i$, the regular momentum $\pi_a$, and the function $g_i$" That's why things were confusing for me. Can I now, think of the coordinates in this way $(q^i,\pi_{a},\phi_{\alpha})$, where $\pi_{a}$'s are regular momenta? Then also $a+\alpha=n$, so we'd have a $2n$-dimensional manifold? Jun 20 at 1:16
• $\uparrow$ Yes. Jun 20 at 1:19