Lagrangian approach to the physical degree of freedom count

Abstract

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.Comment: LaTeX file, no figure

Maxwell-Proca theory: Definition and construction

“…Recall that a first class constraint is one that yields a vanishing Poisson bracket with all of the constraints in the theory, whereas a second class constraint does not. It is worth noting that in [28] an alternative expression to (5) was developed for particle systems, which depends solely on Lagrangian quantities, including the off-shell gauge generators. The result was later on adapted to field theories in [29].…”

Geometric Lagrangian approach to the physical degree of freedom count in field theory

“…Furthermore, it was also shown in Ref. 10 that the Lagrangian formula can be cast in geometric terms such as N − 1 2 (l + g + e) = 1 2 Rank (ι * Ω) ,…”

“…6 The constraint algorithm generates a sequence of sub-manifolds T C =: P 1 ⊇ P 2 ⊇ P 3 ⊇ · · · which are defined by the Lagrangian constraints ϕ ′ s (many of them are actually part of the equations of motion). 6,10 The algorithm must end (if the theory is well-defined) at some final constraint submanifold P := P s = ∅, 1 ≤ s < ∞. Thereby, on P , we have completely consistent Lagrange's equations of motion…”

Exorcising the Ostrogradsky ghost in coupled systems

“…These are, by the converse of Noether's second theorem, in one-to-one correspondence with gauge symmetries, which one can then explicitly obtain [28]. The relation with the degree of freedom counting in the Hamiltonian formalism can be seen by noting that the number of second class Hamiltonian constraints is given by l + g − e and the number of first class Hamiltonian constraints equals e. Also, the total number of gauge identities g equals the number of primary first class constraints in the Hamiltonian analysis; see again [26][27][28][29].…”

“…gauge identities. After the termination of this algorithm, the number of degrees of freedom present in the theory can be computed to be [26][27][28][29] …”