On the derivation of Hamilton's equations
type: Item
instance of: scholarly article
title: On the derivation of Hamilton's equations
zbMATH Open document ID: 0809.70012
DOI: 10.1007/BF00375060
author: Patrizia Pucci
author: James Serrin
published in: Archive for Rational Mechanics and Analysis
publication date: 1995-03-29T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 562429
zbMATH Keywords: time-dependent Lagrangian
zbMATH Keywords: Legendre transform
zbMATH Keywords: variational analysis
reviewed by: Hernan Cendra
MaRDI profile type: Publication
cites work: Optimization and nonsmooth analysis
cites work: Q4085497
cites work: Q5796405
cites work: Critical point theory and Hamiltonian systems
cites work: Precise damping conditions for global asymptotic stability for nonlinear second order systems
cites work: Convex Analysis
cites work: Q4185862
description: scientific article; zbMATH DE number 562429