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How can I apply the Hamiltonian function and Pontryagin's maximum principle in the context of Optimal Control Theory?

I am really struggling to grasp how the Hamiltonian Function and Pontryagin's Maximum Principle work in the context of Optimal Control Theory (Maths for Economics) course. I am given the following ...
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How to add a linear contraint between state variables to a current time Hamiltonian?

Let's say I have an objective function $F$ with state variables $A,B,C$ and relative equations of motion, I can create the current time Hamiltonian with $H_C = F\{A,B,C\}+\alpha * \dot A+\beta * \dot ...