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Prog. Theor. Phys. Vol. 67 No. 6 (1982) pp. 1966-1988

[ Full Text PDF : FREE ACCESS (796K) ]

Poincaré-Cartan Invariant Form and Dynamical Systems with Constraints

Reiji Sugano and Hideki Kamo*

Department of Physics, Osaka City University, Osaka 558
*Research Institute for Atomic Energy, Osaka City University, Osaka 558

(Received February 15, 1982)

Abstract:

A formulation of constrained dynamical systems is developes on the basis of the Poincaré-Cartan invariant form both in the Lagrangian and in the Hamiltonian formalism. The exterior differential form is used in analysis of the equations of motion.
It is shown that by requiring the consistent correspondence of the Lagrangian formalism with the Hamiltonian one, the ambiguity in the choice of Hamiltonian in the Dirac theory can be removed and the total Hamiltonian, the generator of evolution of a constrained system, is uniquely determined, except for arbitrary gauge functions.
The (n×n) singular Hessian matrix Aij = ∂2L/\dotqi\dotqj with the rank n - r, has eigenvectors ταi(α = 1∼r) belonging to the zero eigenvalue. Those ταi are correlated with primary constraints Φα(q,p) in the phase space by ταi = ∂Φα/∂pi. In the vector field which generates evolution of the system, ταi appear accompanying undetermined coefficient functions, in proper response to Φα in the Hamiltonian in the Dirac theory. In the Lagrangian formalism, integrable equations which contain ταi with the undetermined coefficient functions are obtained. Some of the coefficient functions are determined by integrability conditions, but others remain arbitrary which give gauge freedom.
If the rank of Aij is reduced further to n-r-r by taking account of (secondary) constraints χσ, extra eigenvectors τβi exist under mod(χσ). The τβi enter in the formulation on the same footing with ταi. Then the generalized Hamiltonian is given by H0 = H+vαΦα+vβΦβ where τβi = ∂Φβ/pi and H is the canonical Hamiltonian. Φα and Φβ are called “intrinsic constraints” in order to distinguish them from χΣ. It is shown that first class intrinsic constraints are associated with gauge freedom, but those of χσare not. Finally it is remarked that when the first class secondary constraints χΣ appear, the first class intrinsic constraints are not necessarily correct generators of the gauge transformations, but the correct ones can be expressed as linear combinations of the first class ΦA (or ΦB) and χΣ.


URL : http://ptp.ipap.jp/link?PTP/67/1966/
DOI : 10.1143/PTP.67.1966

[ Full Text PDF : FREE ACCESS (796K) ] Citation:

References:

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Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 68 No. 4 (1982) pp. 1377-1393 :
    Poincaré-Cartan Invariant Form and Dynamical Systems with Constraints. II
    Reiji Sugano
  2. Progress of Theoretical Physics Vol. 69 No. 1 (1983) pp. 252-261 :
    On the Relation of First-Class Constraints to Gauge Degrees of Freedom
    Reiji Sugano and Toshiei Kimura
  3. Progress of Theoretical Physics Vol. 69 No. 4 (1983) pp. 1256-1271 :
    On the Choice of Gauge Conditions for Systems with Secondary First Class Constraints
    Reiji Sugano and Toshiei Kimura
  4. Progress of Theoretical Physics Vol. 70 No. 1 (1983) pp. 36-50 :
    Necessary and Sufficient Conditions of Reducibility of Gauge Degrees of Freedom without Using Gauge Conditions
    Reiji Sugano
  5. Progress of Theoretical Physics Vol. 73 No. 4 (1985) pp. 1025-1042 :
    Relation between Generators of Gauge Transformations and Subsidiary Conditions on State Vectors
    Reiji Sugano, Tohru Sohkawa and Toshiei Kimura
  6. Progress of Theoretical Physics Vol. 76 No. 1 (1986) pp. 283-301 :
    Generator of Gauge Transformation in Phase Space and Velocity Phase Space
    Reiji Sugano, Yoshihiko Saito and Toshiei Kimura
  7. Progress of Theoretical Physics Vol. 90 No. 4 (1993) pp. 917-929 :
    More General Momentum Dependent Gauge Transformations
    Yusho Kagraoka
  8. Progress of Theoretical Physics Vol. 122 No. 2 (2009) pp. 323-337 :
    Relativistic Particle in Complex Spacetime
    Takayuki Hori

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