First-class constraint: Difference between revisions

Revision as of 03:28, 30 November 2019

Revision as of 05:55, 30 December 2019

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Before analyzing the Hamiltonian, consider the three constraints,

:<math>

\phi_1varphi_1 = p_\lambda, \quad \phi_2varphi_2 = r^2-R^2, \quad \phi_3varphi_3 = \vec{p}\cdot\vec{r}.

</math>

Note the nontrivial [[Poisson bracket]] structure of the constraints. In particular,

:<math>

\{\phi_2varphi_2, \phi_3varphi_3\} = 2 r^2 \neq 0.

</math>

The above Poisson bracket does not just fail to vanish off-shell, which might be anticipated, but ''even on-shell it is nonzero''. Therefore, {{math| ''φ''₂}} and {{math| ''φ''₃}} are '''second class constraints''' while {{math| ''φ''₁}} is a first class constraint. Note that these constraints satisfy the regularity condition.