Presymplectic lagrangian systems. I : the constraint algorithm and the equivalence theorem

Gotay, Mark J.; Nester, James M.

Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 30 (1979) no. 2, pp. 129-142.

[1] R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin, New York, second edition, 1978. | MR | Zbl

[2] J. Klein, Ann. Inst. Fourier (Grenoble), t. 12, 1962, p. 1 ; Symposia Mathematica XIV (Rome Conference on Symplectic Manifolds), 181, 1973. | MR

[3] D.J. Simms and N.M.J. Woodhouse, Lectures on Geometric Quantization, Lecture Notes in Physics, t. 53, Springer-Verlag, Berlin, 1976. | MR | Zbl

[4] A nice summary is given in P.A.M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monograph Series, t. 2, 1964. Several examples are presented in Hanson, Regge and Teitelboim, Accademia Nazionale dei Lincei (Rome), t. 22, 1976.

[5] M.J. Gotay, J.M. Nester and G. Hinds, Presymplectic Manifolds and the Dirac-Bergmann Theory of Constraints, J. Math. Phys., t. 19, 1978, p. 2388. | MR | Zbl

[6] H.P. Künzle, Ann. Inst. H. Poincaré, t. A 11, 1969, p. 393. | Numdam | MR | Zbl

[7] From the point of view of the constraint algorithm, the homogeneous case is trivial because E ≡ 0 (see section III).

[8] J.M. Nester and M.J. Gotay, Presymplectic Lagrangian Systems II: The Second-Order Equation Problem (in preparation). | Numdam | Zbl

[9] J. Sniatycki, Proc. 13th Biennial Seminar of the Canadian Math. Cong., t. 2, 1972, p. 125. | MR | Zbl

[10] Throughout this paper, we assume for simplicity that all physical systems under consideration are time-independent and that all relevant phasespaces are finite-dimensional; however, all of the theory developed in this paper can be applied when these restrictions are removed with little or no modification. For details concerning the infinite-dimensional case, see refs. [5] and [18].

[11] C. Godbillon, Géométrie Différentielle et Mécunique Analytique, Hermann, Paris, 1969. | MR | Zbl

[12] We herein establish some notation and terminology. All manifolds and maps appearing in this paper are assumed to be C∞. We designate the natural pairing TM x T*M → R by <|>. The symbol i denotes the interior product. Note that if γ is a p-form, and X1, ..., Xp are vectorfields, then i(X1) ... i(Xp)γ = γ(Xp, ... , X1). The symbol « | N » means « restriction to the submanifold N ». If j : N → M is the inclusion, then we denote by γ | N the restriction of γ to N. Given a 2-form Ω on M, we define the « Ω-orthogonal complement » of TN in TM to be TN1 = {Z∈TM such that Ω(Z, Y) = 0 for all Y∈TN}. Furthermore, we define ker Ω = {Y∈TM such that i(Y)Ω = 0 }. If f : M → P is smooth, then we denote by T f or f* the derived mapping TM → TP. We have ker T f = { Y∈TM such that T f(Y) = 0 } .

[13] For another definition of FL (which is logically independent of the almost tangent structure J), see ref. [1].

[14] J.M. Nester, Invariant Derivation of the Euler-Lagrange Equations (in preparation).

[15] We assume that all of the Pl appearing in the algorithm are in fact imbedded submanifolds. Otherwise, one must resort to standard tricks, e. g., work locally where everything is manageable (see Section IV).

[16] In fact, there does not even exist a unique local Hamiltonian formalism corresponding to such a Lagrangian system, as, e. g., with L = 1/4v4 - 1/2v2 .

[17] In the following, TM1/1 denotes the ω1-orthogonal complement (see [12]).

[18] M.J. Gotay, Presymplectic Manifolds, Geometric Constraint Theory and the Dirac-Bergmann Theory of Constraints, Ph. D. Thesis, University of Maryland, 1979.