Primary constraint
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The distinction between primary and secondary constraints is not a very fundamental one. It depends very much on the original Lagrangian which we start off with. Once we have gone over to the Hamiltonian formalism, we can really forget about the distinction between primary and secondary constraints.Lua error in package.lua at line 80: module 'Module:Footnotes/anchor_id_list' not found.
In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion.Lua error in package.lua at line 80: module 'Module:Footnotes/anchor_id_list' not found. A secondary constraint is one that is not primary—in other words it holds when the equations of motion are satisfied, but need not hold if they are not satisfiedLua error in package.lua at line 80: module 'Module:Footnotes/anchor_id_list' not found. The secondary constraints arise from the condition that the primary constraints should be preserved in time. A few authors use more refined terminology, where the non-primary constraints are divided into secondary, tertiary, quaternary, etc. constraints. The secondary constraints arise directly from the condition that the primary constraints are preserved by time, the tertiary constraints arise from the condition that the secondary ones are also preserved by time, and so on. Primary and secondary constraints were introduced by Anderson and BergmannLua error in package.lua at line 80: module 'Module:Footnotes/anchor_id_list' not found. and developed by Dirac.[1]Template:R/superscript[2]Template:R/superscript[3]Template:R/superscriptLua error in package.lua at line 80: module 'Module:Footnotes/anchor_id_list' not found.
The terminology of primary and secondary constraints is confusingly similar to that of first- and second-class constraints. These divisions are independent: both first- and second-class constraints can be either primary or secondary, so this gives altogether four different classes of constraints.
References
- Page Module:Citation/CS1/styles.css has no content.Anderson, James L.; Bergmann, Peter G. (1951). "Constraints in covariant field theories". Physical Review. Series 2. 83 (5): 1018–1025. Bibcode:1951PhRv...83.1018A. doi:10.1103/PhysRev.83.1018. MR 0044382.
- Page Module:Citation/CS1/styles.css has no content.Dirac, Paul A. M. (1964). Lectures on quantum mechanics. Belfer Graduate School of Science Monographs Series. Vol. 2. New York: Belfer Graduate School of Science. ISBN 9780486417134. MR 2220894.
{{cite book}}: ISBN / Date incompatibility (help) 2001 reprint by Dover.
Footnotes
- ^ Page Module:Citation/CS1/styles.css has no content.Dirac, Paul A. M. (1950). "Generalized Hamiltonian dynamics". Canadian Journal of Mathematics. 2: 129–148. doi:10.4153/CJM-1950-012-1. ISSN 0008-414X. MR 0043724. S2CID 119748805.
- ^ Page Module:Citation/CS1/styles.css has no content.Dirac, Paul A. M. (1958). "Generalized Hamiltonian dynamics". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 246 (1246): 326–332. Bibcode:1958RSPSA.246..326D. doi:10.1098/rspa.1958.0141. ISSN 0962-8444. JSTOR 100496. MR 0094205. S2CID 122175789.
- ^ Page Module:Citation/CS1/styles.css has no content.Dirac, Paul A. M. (1958). "The theory of gravitation in Hamiltonian form". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 246 (1246): 333–343. Bibcode:1958RSPSA.246..333D. doi:10.1098/rspa.1958.0142. ISSN 0962-8444. JSTOR 100497. MR 0094206. S2CID 122053391.
Further reading
- Page Module:Citation/CS1/styles.css has no content.Salisbury, D. C. (2006). Peter Bergmann and the invention of constrained Hamiltonian dynamics. arXiv:physics/0608067. Bibcode:2006physics...8067S.
