Prog. Theor. Phys. Vol. 67 No. 6 (1982) pp. 1966-1988
Poincaré-Cartan Invariant Form and Dynamical Systems with Constraints
Department of Physics, Osaka City University, Osaka 558
*Research Institute for Atomic Energy, Osaka City University, Osaka 558
(Received February 15, 1982)
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 =
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.
DOI : 10.1143/PTP.67.1966
- P. A. M. Dirac, in Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, New York, 1964).
R. Cawley, Phys. Rev. Lett. 42 (1979), 413[APS].
R. Cawley, Phys. Rev. D 21 (1980), 2988[APS].
G. R. Allcock, Philos, Trans. R. Soc. London A 279 (1975), 487.
A. Frenkel, Phys. Rev. D 21 (1980), 2986[APS].
M. J. Gotay and J. M. Nester, J. Math. Phys. 19 (1978), 2388[CrossRef].
- M. J. Gotay and J. M. Nester, in Lecture Notes in Phys. 94 (Springer-Verlag, Berlin, 1979), p. 272.
S. Shanmugadhassan, J. Math. Phys. 14 (1973), 677[CrossRef].
- E. C. G. Sudarshan and N. Mukunda, in Classical Mechanics; A Modern Perspective (Wiley, New York, 1974).
- M. J. Gotay and Nester, Ann. Inst. Henri Poincaré 30 (1979), 129; ibid. 32 (1980), 1; in Lecture Notes in Mathematics 775 (Springer-Verlag, Berlin, 1980).
- A. Hanson, T. Regge and C. Teitelboim, in Accademia Nationale dei Lincei, Rome No. 22 (1976).
- R. Abraham and J. E. Marsden, in Foundations of Mechanics (Benjamin, 1978).
- R. Sugano, Preprint OCU 85.
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