First-class constraint: Difference between revisions

Revision as of 17:17, 17 January 2016

Revision as of 17:26, 17 January 2016

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where the last term is the [[Lagrange multiplier]] term enforcing the constraint.

Of course, as indicated, we could have just used different, non-redundant, spherical [[coordinates]] and written it as

:<math>S=\int dt \left[\frac{mR^2}{2}(\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2)+mgR\cos(\theta)\right]</math>

instead, without extra constraints,; but we lookare atconsidering the former coordinatization to illustrate constraints.

The [[conjugate momentum|conjugate momenta]] are given by

:<math>p_x=m\dot{x}</math>, <math>p_y=m\dot{y}</math>, <math>p_z=m\dot{z}</math>, <math>p_\lambda=0</math> .

Note that we can't determine < {{math>\dot|{\lambda{overset|•|''λ''}</math>}}} from the momenta.

The [[Hamiltonian mechanics|Hamiltonian]] is given by

:<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 <math>\dot {\lambda{overset|•|''λ''}}</math> at this stage yet. We are here treating <math>\dot {\lambda{overset|•|''λ''}</math>} 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_1=\dot{\lambda}</math> 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>\dot| {\lambda{overset|•|''λ''}} {{=u_1}} ''u''<sub>1</mathsub>}}.

We have the [[primary constraint]]