Physical degree of freedom count determined by Lagrangian parameters
Keywords:
Constrained Hamiltonian systems, Singular Lagrangians
Abstract
We exhibit a simple motivation for the Díaz-Higuita-Montesinos formula to deduce the number of physical degrees of freedom in singular systems, and for three Lagrangians we illustrate the validity of this formula.
References
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Lagrangians, Theor. Math. Phys. 64, No. 1 (1985) 701-707
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de l’École Polytechnique 8 (1809) 266-344
31. C. Lanczos, The variational principles of mechanics, University of Toronto Press (1970)
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Cambridge University Press (1972) 169-178
J. Math. Phys. 55 (2014) 122901
2. H. J. Rothe, K. D. Rothe, Classical and quantum dynamics of constrained Hamiltonian systems, World
Scientific Lecture Notes in Physics 81, Singapore (2010)
3. P. Lam, J. López-Bonilla, R. López-Vázquez, G. Ovando, Lagrangians: Symmetries, gauge identities,
and first integrals, The SciTech, J. of Sci. & Tech. 3, No. 1 (2014) 54-66
4. P. Lam, J. López-Bonilla, R. López-Vázquez, G. Ovando, On the gauge identities and genuine
constraints of certain Lagrangians, Prespacetime Journal 6, No. 3 (2015) 238-246
5. P. Lam, J. López-Bonilla, R. López-Vázquez, G. Ovando, Matrix method to construct point symmetries
of Lagrangians, Bull. of Kerala Mathematics Assoc. 12, No. 1 (2015) 43-52
6. P. Lam, J. López-Bonilla, R. López-Vázquez, M. Maldonado, On a recent formula to determine the
physical degree of freedom count, Prespacetime Journal 7, No. 9 (2016) 1306-1312
7. L. Rosenfeld, On the quantization of wave fields, Ann. der Phys. 5 (1930) 113-152
8. P. Hanson, T. Regge, C. Teitelboim, Constrained Hamiltonian systems, Accad. Naz. dei Lincei, Rome
(1976)
9. C. A. Hurst, Dirac’s theory of constraints, in ‘Proc. Recent developments in Mathematical Physics’,
Eds. H. Mitter, L. Pitter; Springer-Verlag, Berlin (1987) 18-52
10. P. Bergmann, The canonical formulation of general-relativistic theories: The early years, 1930-1959,
in ‘Einstein and the history of general relativity’, Eds. D. Howard, J. Stachel; Birkhäuser, Boston
(1989) 293-299
11. A. Wipf, Hamilton’s formalism for systems with constraints, in ‘Canonical gravity: From classical to
quantum’, Eds. J. Ehlers, H. Friedrich; Springer-Verlag, Berlin (1994) 22-58
12. M. Henneaux, C. Teitelboim, Quantization of gauge systems, Princeton University Press, NJ (1994)
13. J. Earman, Tracking down gauge: an ode to the constrained Hamiltonian formalism, in ‘Symmetries in
Physics. Philosophical Reflections’, Eds. Katherine Brading, Elena Castellani; Cambridge University
Press (2003) 140-162
14. D. C. Salisbury, Rosenfeld, Bergmann, Dirac and the invention of constrained Hamiltonian dynamics,
arXiv: physics/0701299v1 [physics.hist-ph] 25 Jan 2007
15. D. C. Salisbury, Peter Bergmann and the invention of constrained Hamiltonian dynamics, in ‘Einstein
and the changing worldviews of Physics’, Eds. C. Lehner, J. Renn, M. Schemmel; Einstein Studies 12,
Birkhäuser, Boston (2012) 247-257
16. P. Dirac, Lecture on quantum mechanics, Yeshiva University, New York (1964)
17. M. J. Gotay, J. M. Nester, G. Hinds, Presympletic manifolds and the Dirac-Bergmann theory of
constraints, J. Math. Phys. 19, No. 11 (1978) 2388-2399
18. E. Castellani, Dirac on gauges and constraints, Int. J. Theor. Phys. 43, No. 6 (2004) 1503-1514
19. J. N. Goldberg, Second-class constraints, in ‘On Einstein’s path. Essays in honor of Engelbert
Schucking’, Ed. A. Harvey; Springer-Verlag, New York (1999) 251-256
20. G. R. Allcock, The intrinsic properties of rank and nullity of the Lagrange bracket in the one
dimensional calculus of variations, Phil. Trans. Roy. Soc. London 279, No. 1290 (1975) 487-545
21. R. Cawley, Determination of the Hamiltonian in the presence of constraints, Phys. Rev. Lett. 42, No. 7
(1979) 413-416
22. X. Gracia, J. M. Pons, Gauge generators, Dirac’s conjecture, and degrees of freedom for constrained
systems, Ann. of Phys. 187, No. 2 (1988) 355-368
23. M. E. V. Costa, H. O. Girotti, T. J. M. Simes, Dynamics of gauge systems and Dirac’s conjecture, Phys.
Rev. D32, No. 2 (1985) 405-410
24. A. Cabo, D. Louis-Martínez, On Dirac’s conjecture for Hamiltonian systems with first-and second-class
constraints, Phys. Rev. D42, No. 8 (1990) 2726-2735
25. M. Henneaux, C. Teitelboim, J. Zanelli, Gauge invariance and degree of freedom count, Nucl. Phys.
B332, No. 1 (1990) 169-188
26. M. Havelková, Symmetries of a dynamical system represented by singular Lagrangians, Comm. in
Maths. 20, No. 1 (2012) 23-32
27. G. F. Torres del Castillo, Point symmetries of the Euler-Lagrange equations, Rev. Mex. Fís. 60 (2014)
129-135
28. R. L. Schafir, In defence of the Dirac theory of constraints, J. Phys. A: Math. Gen. 15, No. 7 (1982)
L331-L336
29. V. V. Nesterenko, A. M. Chervyakov, Some properties of constraints in theories with degenerate
Lagrangians, Theor. Math. Phys. 64, No. 1 (1985) 701-707
30. S. D. Poisson, Mémoire sur la variation des constants arbitraries dans les questions de mécanique, Jour.
de l’École Polytechnique 8 (1809) 266-344
31. C. Lanczos, The variational principles of mechanics, University of Toronto Press (1970)
32. C. Lanczos, The Poisson bracket, in ‘Aspects of quantum theory’, Eds. A. Salam, E. P. Wigner;
Cambridge University Press (1972) 169-178
Published
2019-02-02
How to Cite
Bonilla, D. L.-, Hinojosa, A. I.-, Vázquez, R. L.-, & Beltrán, S. V.-. (2019). Physical degree of freedom count determined by Lagrangian parameters. IJO - International Journal of Mathematics (ISSN: 2992-4421 ), 2(01), 01-06. Retrieved from https://www.ijojournals.com/index.php/m/article/view/68
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