zbMATH — the first resource for mathematics

Geometric theory of the equivalence of Lagrangians for constrained systems. (English) Zbl 0588.58020
The authors present a geometric description of Lagrangian systems (§ 1). The notations and the basic definitions are given in § 2. If Q is a differentiable manifold modeling a mechanical system, let TQ be the tangent bundle (\(\pi\) : TQ\(\to Q)\). A function \(L\in C^{\infty}(TQ)\) is called a regular Lagrangian if the form \(\omega_ L=D^*_ L\omega_ 0\in Z^ 2(TQ)\) has maximal rank, where \(D_ L: TQ\to T^*Q\) is the Legendre transform and \(\omega_ 0\) is the canonical symplectic 2-form on \(T^*Q\). In § 3 some results on the tangent bundle are developed following M. J. Gotay and J. M. Nester [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 30, 129-142 (1979; Zbl 0414.58015)]. The gauge equivalence for singular Lagrangians is defined in § 4. So, one has \(L_ 1\sim L_ 2\) if there is a 1-form \(\alpha \in Z^ 1(Q)\) such that \(L_ 2=L_ 1+{\hat \alpha}\) up to a constant, where \({\hat \alpha}\)(u)\(=\alpha_{\pi (u)}(u)\), \({\hat \alpha}\in C^ 1(TQ)\). The main theorems refer to the existence of a second order differential equation satisfying the dynamical equations on the final constraint submanifold. In § 5 and § 6 some examples of Lagrangians are given and final remarks are made.
Reviewer: M.Tarina

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H03 Lagrange’s equations
53C80 Applications of global differential geometry to the sciences
Full Text: DOI