Lagrangian approach to the physical degree of freedom count

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American Institute of Physics (AIP)
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In this paper, we present a Lagrangian method that allows the physical degree of freedom count for any Lagrangian system without having to perform neither Dirac nor covariant canonical analyses. The essence of our method is to establish a map between the relevant Lagrangian parameters of the current approach and the Hamiltonian parameters that enter in the formula for the counting of the physical degrees of freedom as is given in Dirac’s method. Once the map is obtained, the usual Hamiltonian formula for the counting can be expressed in terms of Lagrangian parameters only, and therefore we can remain in the Lagrangian side without having to go to the Hamiltonian one. Using the map, it is also possible to count the number of first and second-class constraints within the Lagrangian formalism only. For the sake of completeness, the geometric structure underlying the current approach—developed for systems with a finite number of degrees of freedom—is uncovered with the help of the covariant canonical formalism. Finally, the method is illustrated in several examples, including the relativistic free particle.
Gauge fixing, Partial differential equations, Lagrangian mechanics, Constraint algorithm, Operator theory, Hamiltonian mechanics, Philosophy of mathematics, Gauge symmetry, Differential geometry, Calculus of variations
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Díaz, B., Higuita, D. & Montesinos, M. (2014). Lagrangian approach to the physical degree of freedom count. Journal of Mathematical Physics, 55(12), 122901.