First-class constraint: Difference between revisions

Revision as of 21:26, 17 January 2016

Revision as of 21:55, 17 January 2016

Line 13:

:<math> f_i(x)=0, </math>

for ''n'' smooth functions

:<math>\{ f_i \}_{i= 1}^n</math>

These will only be defined [[chart (topology)|chartwise]] in general. Suppose that everywhere on the constrained set, the ''n'' derivatives of the ''n'' functions are all [[linearly independent]] and also that the [[Poisson bracket]]s

:<math>\{f_i,f_j\}</math>

and

:<math>\{f_i,H\}</math>

all vanish on the constrained subspace.

all vanish on the constrained subspace. This means we can write

:<math>\{f_i,f_j\}=\sum_k c_{ij}^k f_k</math>

for some smooth functions <math>c_{ij}^k</math> −−there is a theorem showing this; and

for some smooth functions

:<math>c_{ij}^k</math>

(there is a theorem showing this) and

:<math>\{f_i,H\}=\sum_j v_i^j f_j</math>

for some smooth functions <math>v_i^j</math>.

for some smooth functions

:<math>v_i^j</math>.

This can be done globally, using a [[partition of unity]]. Then, we say we have an irreducible '''first-class constraint''' (''irreducible'' here is in a different sense from that used in [[representation theory]]).