Talk:Second class constraints

The author calls $r^{2}-R^{2}=0$ a secondary constraint, and calls the following ones with different names, thus giving a wrong impression of what is really meant by a secondary class constraint. Besides, I dont think it is clear how those constraints are acquired.

Sticking to this example it is necessary to clarify why we need all 4 of the constraints, and why they are all classified as secondary class constraints.

--Zjappar 22:37, 5 August 2006 (UTC)

The problem you're having is that you (quite understandably) are confusing "secondary constraint" with "second class constraint" which are completely distinct concepts. A secondary constraint is a constraint that one finds by demanding, on the grounds of consistency, that the primary constraints (whatever constraints you started with) have vanishing time derivative. So not only do you demand that your constraint \phi = 0 you also demand that \dot{\phi} = 0. Sometimes this will lead to a new constraint (frequently not). Constraints arrived in this manner are called "secondary constraints". The distinction between primary constraints (whatever constraints you start with in the Hamiltonian formalism) or secondary constraints or tertiary constraints, etc. is largely artificial and unimportant. A dymanical quantity is called first class if its poisson bracket with all constraints vanishes on-shell, and second-class if its poisson bracket does not vanish with all constraints on-shell, then it is called second class. Hence, a second class constraint has nonvanishing poisson bracket with at least one nother constraint. The distinction between primary and secondary constraints is very important. Perhaps the Dirac bracket article I wrote will clarify. Steve Avery 17:11, 4 December 2007 (UTC)