On the derivation of Hamilton's equations
Let \({\mathcal L}(t, u, u')\) be a time-dependent Lagrangian such that: (i) The function \({\mathcal L}(t, u, p)\) is defined and of class \(C^ 1\) in a domain \(I \times \Omega\), where \(I\) is an open set of \(\mathbb{R}^{1 + N}\) and \(\Omega\) is an open set of \(\mathbb{R}^ N\); (ii) The function \({\mathcal L}(t, u, \cdot)\) is strictly convex in the sense of Weierstrass for every fixed \((t,u) \in I\) (this means that \({\mathcal L}(t, u, p) - {\mathcal L}(t, u, p_ 0) > ({\mathcal L}(t, u, p_ 0), p - p_ 0)\) for all \(p, p_ 0 \in \Omega\), \(p \neq p_ 0\); here \(( , )\) is the inner product in \(\mathbb{R}^ n\)). Under these conditions, the authors show that the function \({\mathcal L}_ p(t, u, \cdot)\), for fixed \((t, u) \in I\), is a homeomorphism and that the inverse \(\mathcal P\) of \({\mathcal L}(t, u, \cdot)\) is continuous as a function of \((t, u, v)\). Then they introduce the Legendre transform \(\mathcal H\) of \(\mathcal L\) as follows. Let \(H(t, u, p) = ({\mathcal L}_ p (t, u, p),p) - {\mathcal L}(t, u, p)\). Then define \(\mathcal H\) by \({\mathcal H}(t, u, v) = H(t, u, p)\), where \(p = {\mathcal P}(t, u, v)\), \(v = {\mathcal L}_ p(t, u, p)\). After this they show that \(\mathcal H\) is continuously differentiable with respect to \(v\) and that for each \((t,u) \in I\) the function \({\mathcal H}(t, u, \cdot)\) is strictly convex in the sense of Weierstrass. They also show how to make Hamilton's equations and conservation of energy in this framework. The paper generalizes the definition and fundamental properties of the Legendre transform which are usually treated under the \(C^ 2\)-differentiability assumption. Thus it can be applied to several cases of interest in the field of variational analysis. I think it is interesting.
scientific article; zbMATH DE number 562429