Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics
In a recent paper [J. Phys. A, Math. Gen. 22, 3809-3820 (1989)], \textit{M. de Leon} and \textit{E. A. Lacomba} have discussed a generalization to higher-order mechanics of Tulczyjew's description of Lagrangian and Hamiltonian systems via special symplectic manifolds. For a Lagrangian function L living on \(T^ nM\subset TT^{n-1}M\), the corresponding dynamics is somehow defined as a Lagrangian submanifold of \(TT^*T^{n- 1}M\). It is true that the defining relations of this submanifold point to the usual Euler-Lagrange equations, but as the author of the present note rightly observes, such equations, being of order 2n, should properly be obtained as a submanifold of \(T^{2n}M\). The way in which the author achieves this goal goes as follows: he first lets dL determine a Lagrangian submanifold \(S_ L\) on \(TT^{n-1}T^*M\) and subsequently obtains the desired submanifold \({\mathcal E}_ L\subset T^{2n}M\) as the projection into \(T^ nTT^{n-1}M\) of \(T^ nS_ L\cap T^{n+1}T^{n- 1}T^*M\). An explicit coordinate calculation illustrates that the constraint equations of \({\mathcal E}_ L\) are indeed the Euler-Lagrange equations.
scientific article; zbMATH DE number 4147151
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