Lagrangian Noether symmetries as canonical transformations (Q2762021)

Let \(Q\) be a manifold with local coordinates \(q^i\), and \(TQ\) its tangent bundle; consider a Lagrangian \(L\) defined on \(TQ\). The authors consider the case where the Hessian \(\partial^2 L / (\partial {\dot q}^i \partial {\dot q}^j)\) is allowed to be not invertible (singular Lagrangians). An infinitesimal Noether symmetry of \(L\) is a vector field \(X = \xi^i \partial / \partial q^i\) in \(J^n (Q \times \mathbb{R})\) (the \(n\)-th jet space over \(Q \times \mathbb{R}\); here \(n\) is arbitrary but finite) such that under this \(L\) is mapped to \(L + \delta L\) with \(\delta L = d F / d t\) for some smooth function \(F : J^n (Q \times \mathbb{R}) \to \mathbb{R}\). With \(D_t\) being the total time derivative operator in \(J^n (Q \times \mathbb{R})\) and \(E_i [L]\) the Euler-Lagrange equation for \(L\), this can also be rewritten as \(\delta L = E_i [L] \xi^i + D_t G = 0\), where \(G\) is the associated conserved quantity \(G := (\partial L / \partial {\dot q}^i) \xi^i - F\). NEWLINENEWLINENEWLINEWhen dealing with singular theories, one can resort to the Dirac formalism: once a Hamiltonian \(H (p,q)\) and a complete set of independent primary constraints \(\varphi_\mu\) are obtained from \(L\), one introduces Lagrange multipliers \(\lambda^\mu\) and considers \( L_c := p_i {\dot q}^i - H (p,q) - \lambda^\mu \varphi_\mu (p,q) \). In a previous paper [the authors, ibid. A 15, No. 29, 4681-4721 (2000; Zbl 0995.70016)] the authors have shown that a function \(G_c\) generates a Noether symmetry \(X_c = (\delta_c q)^i (\partial / \partial q^i) + (\delta_c p)_i (\partial / \partial p_i)\) according to \(\delta_c q = \{ q, G_c \}\), \(\delta_c p = \{p , G_c \}\) iff \(D_t G_c + \{ G_c , H + \lambda^\mu \varphi_\mu \} = C^\mu \varphi_\mu\) (the proof is reproduced in the appendix of the present paper). NEWLINENEWLINENEWLINEOne could wonder if Noether symmetries of \(L_c\) are still Noether symmetries for \(L\); this paper answers positively and gives an application to the regular case as well as applications to singular cases.