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)) $$ S=\int\mathrm{d}^4xN\sqrt{\gamma}\left[\frac{1}{2\kappa^2}\left(^{(3)}R+K_{ij}K^{ij}-K^2\right)+\frac12[(\Pi_\phi)^2-\partial^i\phi\partial_i\phi]-V(\phi)\right] $$ 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 $$(\Pi_\phi)^2=\dfrac{1}{N^2}\left(\dot{\phi}-N^i\partial_i\phi\right)^2$$ both $N$ and $N^i$ show up, which seems to be a little different from the case in method of Lagrange multipliers I've knwon (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?

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?

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)) $$ S=\int\mathrm{d}^4xN\sqrt{\gamma}\left[\frac{1}{2\kappa^2}\left(^{(3)}R+K_{ij}K^{ij}-K^2\right)+\frac12[(\Pi_\phi)^2-\partial^i\phi\partial_i\phi]-V(\phi)\right] $$ 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 $(\Pi_\phi)^2=\dfrac{1}{N^2}\left(\dot{\phi}-N^i\partial_i\phi\right)^2$ both $N$ and $N^i$ show up, which seems to be a little different from the case in method of Lagrange multipliers I've knwon (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/pdf/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?

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)) $$ S=\int\mathrm{d}^4xN\sqrt{\gamma}\left[\frac{1}{2\kappa^2}\left(^{(3)}R+K_{ij}K^{ij}-K^2\right)+\frac12[(\Pi_\phi)^2-\partial^i\phi\partial_i\phi]-V(\phi)\right] $$ 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 $$(\Pi_\phi)^2=\dfrac{1}{N^2}\left(\dot{\phi}-N^i\partial_i\phi\right)^2$$ both $N$ and $N^i$ show up, which seems to be a little different from the case in method of Lagrange multipliers I've knwon (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?