Identify a Hamiltonian system consistent or not?

Question

I'm sorry if my question is too classic and basic. As Dirac-Bergmann algorithm for Hamiltonian formalism, I find out that a Hamiltonian system is inconsistent if Poisson bracket of primary constraints and Hamiltonian $$\{f_{i},H\}=1\approx 0$$ If the Poisson brackets are equal to $0$ or produce new constraints, the Hamiltonian system is consistent. Am I right about them? Working on a trivial example, such as: $$H=\frac{1}{2}(p^2+x^2)\\f=p^2+x^2+x^4$$ I take Poisson bracket between them and obtain $4x^3p$. So this is consistent? Because there is no obvious result as $1$ appearing.

Answer 1

I) In OP's last example, one gets formally a secondary constraint $x^3p\approx 0$.

However, assuming that $x$ and $p$ are real variables in the phase space $M:=\mathbb{R}^2$, then the primary constraint $$f(x,p)~:=~ p^2+x^2+x^4 ~\approx~ 0 \tag{A}$$ by itself implies that the constraint constrained submanifold $C\subset M$ is just the origin: $$ C~:=~f^{-1}(\{0\})~=~\{(0,0)\}, \tag{B}$$

i.e. $x\approx 0\approx p$. So all dynamics are killed/frozen, and the secondary constraint is automatically satisfied. In summary, OP's last example is a consistent but empty/trivial theory with no DOF.

II) That being said, let us rush to add that the constraint (A) does not fulfill a standard regularity condition, namely that the gradient $\vec{\nabla} f$ should not vanish on the constrained submanifold $C$, cf. e.g. my Phys.SE answer here. In fact the gradient (A) does vanish on $C$.

In general, it is much more demanding to perform the Dirac-Bergmann analysis for non-regular constraints because many standard results from differential geometry does not hold.