To solve the following optimization problem, $$ \max _{\left\{\ell_t\right\}} \int_0^{\infty} e^{-\rho t} u\left(c_t, N_t\right) d t $$ subject to $$ \dot{N}_t =\left(\bar{b}\ell_t-\delta\right) N_t, \\ c_t =w_t\left(1-\ell_t\right) + \varphi \dot{N_t}-\tau_t,\\ $$ where $\ell_t,c_t$ are control variables and $N_t$ is state variable.
I combined the two budget constraints, that is, $$ \dot{N}_t = \frac{1}{1-\varphi}[w_t\left(1-\ell_t\right)+\left(\bar{b}\ell_t-\delta\right) N_t - c_t-\tau_t]. $$ Hence my current Hamiltonian is $$ \mathcal{H}=u\left(c_t, N_t\right)+v_t \frac{1}{1-\varphi}[w_t\left(1-\ell_t\right)+\left(\bar{b}\ell_t-\delta\right) N_t - c_t-\tau_t]. $$ Is that correct?