Poincare-Cartan Invariant Form and Dynamical Systems with Constraints

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概要

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• A formulation of constrained dynamical systems is developed on the basis of the Poincare-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 A_<ij>=∂^2L/q^^.^iq^^.^j with the rank n-r, has eigenvectors τ_α^i(α=1~r) belonging to the zero eigenvalue. Those τ_α^i are correlated with primary constraints Φ_α(q,p) in the phase space by τ_α^i=∂Φ_α/∂p_i. In the vector field which generates evolution of the system, τ_α^i appear accompanying undetermined coefficient functions, in proper response to Φ_α in the Hamiltonian in the Dirac theory. In the Lagrangian formalism, integrable equations which contain τ_α^i 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 A_<ij> is reduced further to n-r-r^^- by taking account of (secondary) constraints χ_σ, extra eigenvectors τ^^-_β^i (β=1~r^^-) exist under mode(χ_σ). The τ^^-_β^i enter in the formulation on the same footing with τ_α^i. Then the generalized Hamiltonian is given by H_g=H+v^αΦ_α+v^^-^βΦ^^-_β where τ^^-_β^i=∂Φ^^-_β/∂p_i and H is the canonical Hamiltonian. Φ_α and Φ_β are called "intrinsic constraints are associated with gauge freedom, but those of χ_σ are not. Finally it is remarked that when the first class secondary constraints χ_Σ 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 Φ_A(or Φ^^-_B) and χ_Σ.

• 理論物理学刊行会の論文
• 1982-06-25

著者

• Sugano Reiji

  Department Of Physics Kyoto University

• Kamo Hideki

  Reseach Institute For Atomic Energy Osaka City University

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