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The author calls 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)
I approach this material from a slightly different perspective (see Dirac bracket, I would make a primary constraint and drop the Lagrange multiplier), but I believe that this article is slightly incorrect in a couple subtle ways:
1) Relating what has been done here to the formalism I am familiar with, corresponds to , that is the arbitrary coefficient of the constraint added to the naive hamiltonian. I agree, then, with the Hamiltonian, and that we have a primary constraint . The primary constraint, indeed gives a secondary constraint, . We then should add this secondary constraint into the Hamiltonian with an arbitrary coefficient . Consistency conditions result in a "tertiary" constraint . We need to then add it, too, into the Hamiltonian with an arbitrary coefficient . Now, the consistency conditions result in no further constraints, but they do fix some of the u's. If I calculate correctly, we find that
and
Note that u_1 or never got fixed. This corresponds to the fact that is in fact a first class constraint. The gauge freedom associated with it, is the freedom to pick . So, in fact, this Lagrangian leads to a Hamiltonian system with one first class constraint, and two second class constraints. This distinction between new constraints and consistency conditions that fix the u's is important when one wants to compute Dirac brackets (if one wants to quantize the system).
2) In the literature that I have encountered, people would call secondary, tertiary, and quaternary constraints all "secondary constraints". This is more or less just an issue of semantics.
3) I am not sure what "off-shell constraint" is supposed to mean. If anything, I would call that an "on-shell" constraint since it need only be satisfied when the equations of motion are satisfied. If the constraint had to be satisfied off-shell too, then we should drop it from the Hamiltonian.
If no one responds within a few days, I will begin fixing the above problems. I think the article needs some more drastic changes too, but I will propose more changes here before making any more serious alterations. Steve Avery ( talk) 19:03, 6 December 2007 (UTC)
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The author calls 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)
I approach this material from a slightly different perspective (see Dirac bracket, I would make a primary constraint and drop the Lagrange multiplier), but I believe that this article is slightly incorrect in a couple subtle ways:
1) Relating what has been done here to the formalism I am familiar with, corresponds to , that is the arbitrary coefficient of the constraint added to the naive hamiltonian. I agree, then, with the Hamiltonian, and that we have a primary constraint . The primary constraint, indeed gives a secondary constraint, . We then should add this secondary constraint into the Hamiltonian with an arbitrary coefficient . Consistency conditions result in a "tertiary" constraint . We need to then add it, too, into the Hamiltonian with an arbitrary coefficient . Now, the consistency conditions result in no further constraints, but they do fix some of the u's. If I calculate correctly, we find that
and
Note that u_1 or never got fixed. This corresponds to the fact that is in fact a first class constraint. The gauge freedom associated with it, is the freedom to pick . So, in fact, this Lagrangian leads to a Hamiltonian system with one first class constraint, and two second class constraints. This distinction between new constraints and consistency conditions that fix the u's is important when one wants to compute Dirac brackets (if one wants to quantize the system).
2) In the literature that I have encountered, people would call secondary, tertiary, and quaternary constraints all "secondary constraints". This is more or less just an issue of semantics.
3) I am not sure what "off-shell constraint" is supposed to mean. If anything, I would call that an "on-shell" constraint since it need only be satisfied when the equations of motion are satisfied. If the constraint had to be satisfied off-shell too, then we should drop it from the Hamiltonian.
If no one responds within a few days, I will begin fixing the above problems. I think the article needs some more drastic changes too, but I will propose more changes here before making any more serious alterations. Steve Avery ( talk) 19:03, 6 December 2007 (UTC)