# The consistency conditions of constrained Hamiltonian systems

I am studying the Hamiltonian description of a constrained system. There are some questions puzzled me for days, which I have been stuck on it. From the lagrangian, we can obtain the primary constraints, and for the consistency conditions, we have to make thse primary constraints preserved in time, which lead us to some further constraints.

It is clear that $$\{\phi^{(1)}_i,H_c\}+u^{(1)}_j\{\phi^{(1)}_i, \phi^{(1)}_j\}=0 \qquad (1)$$ yeilds the secondary constraints. But what equation would yeild the tertiary constraint? Is it $$\{\phi^{(2)}_a,H_c\}+u^{(1)}_j\{\phi^{(2)}_a, \phi^{(1)}_j\}+u^{(2)}_b\{\phi^{(2)}_a, \phi^{(2)}_b\}=0 \qquad (2)$$ or just $$\{\phi^{(2)}_a,H_c\}+u^{(1)}_j\{\phi^{(2)}_a, \phi^{(1)}_j\}=0 \qquad ? \quad(3)$$ Namely, should we take the secondary constraint into account, when move from the second stage to the third stage. (Here $i,j$ denote the all the dimensions of primary constraints and $a,b$ denote the dimensions of secondary constraints.)

And when should this procedure be terminated? Should we stop or move on, when some constraints don't commute with each other? If we move on, what are the next stage constraints? Is it the parts without the terms involving undetermined $u$s? For example, in Eq.(2), if
$\{\phi^{(2)}_a, \phi^{(1)}_j\}\neq 0$, can it still yeild the tertiary constraints, $\{\phi^{(2)}_a,H_c\}=0$ ?

And which Hamiltonian should we use to derive the Hamilton equations, the Hamiltonian with only primary constraints or the one with all constraints?

I am trying to make my questions clearer, so the description is lengthy. Thanks!

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Do you read this book: Gravitaion and gauge symmetries By m.blagojevic? In the chapter5 i got my answer to this question.if i encounter a better text,i will share it with you. Also we can contact with email or telegram. Good luck – Mja Jun 28 at 21:04
Henneaux and Teitelboim say it is (3) in their book Quantization of Gauge Systems in section 1.1.5, while discussing secondary constraints... – Alex Nelson Jun 28 at 21:46