Lagrange Multipliers in ADM formalism

Question

I'm following this lecture notes (https://javierrubioblog.com/wp-content/uploads/2017/08/adm1.pdf) on ADM formalism. After getting action as (see Eq.(5.41)) $$ S=\int\mathrm{d}^4xN\sqrt{\gamma}\left[\frac{1}{2\kappa^2}\left(^{(3)}R+K_{ij}K^{ij}-K^2\right)+\frac12[(\Pi_\phi)^2-\partial^i\phi\partial_i\phi]-V(\phi)\right]\tag{5.41} $$ it says that due to the missing of (time) derivatives of $N$ and $N^i$, these 2 variables can be considered to be Lagrange multipliers. I don't understand what does this mean, since that in $$(\Pi_\phi)^2=\dfrac{1}{N^2}\left(\dot{\phi}-N^i\partial_i\phi\right)^2\tag{5.40}$$ both $N$ and $N^i$ show up, which seems to be a little different from the case in method of Lagrange multipliers I've known (For example $$L(x,y,\lambda_1,\lambda_2)=f(x,y)+\lambda_1g(x,y)+\lambda_2h(x,y),$$ and there is no term like $\lambda_1\lambda_2\varphi(x,y)$). So I don't understand how to get Eq.(5.43) and Eq.(5.44).

I'm also looking at appendix C of (https://arxiv.org/abs/0907.5424), getting the same problem on deriving Eq.(A.146) and Eq.(A.147) from Eq.(A.143).

Could you please show me how to do this kind of calculation with some details?

Answer 1

1. On one hand in the Lagrangian formalism, the lapse and shift functions $(N,N_i)$ are given in terms of the $g_{0\mu}$-components, and vice-versa, cf. eq. (5.10). $(N,N_i)$ may appear non-linearly in the Lagrangian density. They are auxiliary variables, since their time-derivatives do not appear in the Lagrangian density.

2. On the other hand in the Hamiltonian formalism, the canonical momenta $(\pi_0,\pi^i)$ for $(N,N_i)$ vanish due to primary constraints. $(N,N_i)$ appear linearly in the Hamiltonian Lagrangian density, and are therefore Lagrange multipliers, in fact for the secondary constraints, aka. 1 Hamiltonian + 3 momentum constraints, cf. e.g. my Phys.SE answer here.

Answer 2

The idea of the Hamiltonian formalism to express everything in terms of the fields $\phi$ and momenta $\Pi$ and use those as your fundamental variables. Once you do that, the Hamiltonian has the schematic form $$ \mathcal{H} = N C(\phi, \Pi_\phi, h_{ij}, \pi_{ij}) + N_i C^i(\phi, \Pi_\phi, h_{ij}, \pi_{ij}) $$ where $h_{ij}$ and $\pi_{ij}$ are the spatial metric and its momentum respectively, and where the constraints $C$ and $C_i$ do not depend on $N$ and $N_i$. (Again, remember we are thinking of $\Pi_\phi$ as a fundamental variable here, so you should't rewrite it in terms of $\partial \phi$).

In this form, it's clear that $N$ and $N_i$ are Lagrange multipliers, enforcing the (first-class) constraints $C=C^i=0$.