Replace costate with momentum for Pontryagin's principle

Question

According to the Maximum principle, we know that there exists a costate $\hat{w}:[0,t]\to\mathbb{R}$ such that $\dot{x}=H_{\hat{w}}(x,\hat{w},\alpha)$ and $\dot{\hat{w}}=-H_x(x,\hat{w},\alpha)$ and $H(x,\hat{w},\alpha^*)=\max_{\alpha}H(x,\hat{w},\alpha)$.
Now suppose for some optimal system we want to determine time-optimal control. But we only know Hamiltonian $H(x,p,\alpha)$ with momentum $p$ for that control system. Can we use any modified version of the Maximum principle to obtain time-optimal?
More specifically, if we exchange the role of costate with momentum then is Pontryagin's maximum principle valid?