First-class constraint: Difference between revisions

Revision as of 17:37, 17 January 2016

Revision as of 19:36, 17 January 2016

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:<math>H=\vec{p}\cdot\dot{\vec{r}}+p_\lambda \dot{\lambda}-L=\frac{p^2}{2m}+p_\lambda \dot{\lambda}+mgz-\frac{\lambda}{2}(r^2-R^2)</math>.

We cannot eliminate {{overset|•|''λ''}} at this stage yet. We are here treating {{overset|•|''λ''}} as a shorthand for a function of the [[symplectic manifold|symplectic space]] which we have yet to determine and ''not'' as an independent variable. For notational consistency, define < {{math| ''u''<sub>u_1=\dot{\lambda}1</mathsub> {{=}} {{overset|•|''λ''}} }} from now on. The above Hamiltonian with the {{math|''p''_''λ''}} term is the "naive Hamiltonian". Note that since, on-shell, the constraint must be satisfied, one cannot distinguish, on-shell, between the naive Hamiltonian and the above Hamiltonian with the undetermined coefficient, {{math| {{overset|•|''λ''}} {{=}} ''u''₁}}.

We have the [[primary constraint]]

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From this consistency condition, we immediately get the [[First class constraints#Constrained Hamiltonian dynamics from a Lagrangian gauge theory|secondary constraint]]

:{{math|''r''²−''R''²{{=}}0}} .