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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

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