Hamilton'S Principle and Noether's Theorem

Generalized Noether's theorem. I. theory

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

On integral principles for nonholonomic systems

This work is primarily concerned with various aspects of Lagrangian and Hamiltonian mechanics. These different aspects are related in a somewhat complicated way, which is clarified by the introduction of the unifying concept of a jet bundle. Here, bundles are first considered in some generality and then jet bundles are introduced and their bundle structure investigated especially with a view to formulating Lagrangian and Hamiltonian mechanics invariantly. The second chapter is devoted to a discussion of the two Schouten brackets and it is explained why each is of importance in mechanics. The symmetric Schouten bracket enables the notion of a killing tensor to be defined on a Riemannian or pseudo-Riemannian manifold. A theorem is proved which gives an upper bound on the dimension of the space of killing tensors of a fixed rank. In chapter 3 Hamilton-Jacobi theory and a version of Noether's theorem are presented from a m o d e m viewpoint. Then conditions are obtained which entail the existence of a constant of motion which is polynomial in momenta. These conditions are used to construct several classical Hamiltonians with hidden symmetries - the usage of the latter term is briefly justified. Most importantly, all systems with two degrees of freedom which have a quadratic integral independent of the Hamiltonian are characterized. In chapter 4 the geometrical properties of TM and the closely related space J1 (IR,M) are investigated. It is shown that J1(IR,M) has an intrinsically defined 1-1 tensor in analogy to TM. The behaviour of these tensors under diffeomorphisms is investigated. The chapter concludes with a discussion of the various notions of symmetry in Lagrangian theory. Chapter 5 begins with a review of symplectic geometry and continues with the definition and examination of contact manifolds in the same spirit as symplectic geometry. In particular, contact diffeomorphisms are described on the space J1(IR,M). Finally, two theorems are given which endeavour to explain the general principle enabling the Lie algebra of a subalgebra of vector fields to be transferred to a collection of forms.

An improved derivation involving tensor analysis and covarient differentiation, as well as Hamilton's principle and a realization of Noether's theorem, is given for the energy corollary of the basic shell dynamic equations for a thin shell of arbitrary shape, and with position-dependent elastic modulus E, Poisson's ratio v, and shell thickness h. This corollary states that the sum of (i) the time derivative of an energy density and (ii) the divergence (within the shell middle surface) of a structural intensity vector is identically zero in the absence of external sound radiation and internal damping. The requirement that the intensity transform as a two-dimensional tensor removes the intrinsic ambiguity (with which intensity component does one associate a double derivative term?) that the author had previously reported in an analysis of shells of revolution. The compact tensorial form of this “Poynting's theorem” is seemingly simple, although its explicit unraveling for specific cases can lead to cumbersome expressions. The two tensorial groupings in the structural intensity reduce in the flat plate case to (i) energy flux associated with in-plane (e.g., stretching) motions and (ii) energy flux associated with bending (flexural) motions. Further clarifications of the derived expressions are found in the limiting cases of cylindrical and spherical shells, and in appropriate “high-” and “low-frequency” limits. The latter part of the paper discusses how the corollary may be extended (although with some approximations) to incorporate fluid loading and sound radiation. [Work supported by ONR.]

A concise but rigorous treatment of variational techniques, focussing primarily on Lagrangian and Hamiltonian systems, this book is ideal for physics, engineering and mathematics students. The book begins by applying Lagrange's equations to a number of mechanical systems. It introduces the concepts of generalized coordinates and generalized momentum. Following this the book turns to the calculus of variations to derive the Euler–Lagrange equations. It introduces Hamilton's principle and uses this throughout the book to derive further results. The Hamiltonian, Hamilton's equations, canonical transformations, Poisson brackets and Hamilton–Jacobi theory are considered next. The book concludes by discussing continuous Lagrangians and Hamiltonians and how they are related to field theory. Written in clear, simple language and featuring numerous worked examples and exercises to help students master the material, this book is a valuable supplement to courses in mechanics.

On the coupling of Hamilton's principle and thermodynamic extremal principles

Existence of a variational principle for PDEs with symmetries and current conservation

Noether Theorem and Its Inverse Theorem in Canonical Formalism for Nonholonomic Nonconservative Singular System

Symmetries and first integrals of time-dependent higher-order constrained systems

The procedure describes the derivation of boundary integral equations for surface acoustic waves propagating under periodic metal strip gratings with piezoelectric films. It takes into account the electrical and mechanical perturbations, including the effects of mass loading caused by the gratings with an arbitrary shape. First, an integral equation is derived with line integrals on the boundaries within one period. This derivation is based on Hamilton's principle and uses Lagrange's method of multipliers to alleviate the continuous conditions of the displacement and the electric potential on the boundaries. Second, boundary integral equations corresponding to each substrate, piezoelectric film, metal strip, and free space region are obtained from the integral equation using the Rayleigh-Ritz method for admissible functions. With this procedure, it is not necessary to make any assumptions for separation of the boundary conditions between two neighboring regions. Consequently, we clarify the theoretical basis for the analytical procedure using boundary integral equations for longitudinal LSAW modes.

In the present work, we formulate a generalization of the Noether Theorem for action-dependent Lagrangian functions. The Noether's theorem is one of the most important theorems for physics. It is well known that all conservation laws, \textrm{e.g.}, conservation of energy and momentum, are directly related to the invariance of the action under a family of transformations. However, the classical Noether theorem cannot be applied to study non-conservative systems because it is not possible to formulate physically meaningful Lagrangian functions for this kind of systems in the classical calculus of variation. On the other hand, recently it was shown that an Action Principle with action-dependent Lagrangian functions provides physically meaningful Lagrangian functions for a huge variety of non-conservative systems (classical and quantum). Consequently, the generalized Noether Theorem we present enable us to investigate conservation laws of non-conservative systems. In order to illustrate the potential of application, we consider three examples of dissipative systems and we analyze the conservation laws related to spacetime transformations and internal symmetries.

This paper presents a Hamilton's principle for distributed control of infinite-dimensional systems modeled by a distributed form of the Euler-Lagrange method. The distributed systems are governed by a system of linear partial differential equations in space and time. A generalized potential energy expression is developed that can capture most physical systems including those systems that have no spatial distribution. The Hamilton's principle is applied to derive distributed feedback control methods without resorting to the standard weak-form discretization approach to convert an infinite-dimensional systems to a finite-dimensional systems. It can be shown by the principle of least action that the distributed control synthesized by the Hamilton's principle is a minimum-norm control. A model-reference adaptive control framework is developed for distributed Lagrangian systems in the presence of uncertainty. The theory is demonstrated by an application of adaptive flutter suppression control of a flexible aircraft wing.

SUMMARYAn alternative spatial semi‐discretization of dynamic contact based on a modified Hamilton's principle is proposed. The modified Hamilton's principle uses displacement, velocity and momentum as variables, which allows their independent spatial discretization. Along with a local static condensation for velocity and momentum, it leads to an approach with a hybrid‐mixed consistent mass matrix. An attractive feature of such a formulation is the possibility to construct hybrid singular mass matrices with zero components at those nodes where contact is collocated. This improves numerical stability of the semi‐discrete problem: the differential index of the underlying differential‐algebraic system is reduced from 3 to 1, and spurious oscillations in the contact pressure, which are commonly reported for formulations with Lagrange multipliers, are significantly reduced. Results of numerical experiments for truss and Timoshenko beam elements are discussed. In addition, the properties of the novel discretization scheme for an unconstrained dynamic problem are assessed by a dispersion analysis.Copyright © 2013 John Wiley & Sons, Ltd.

We argue that Hamiltonian mechanics is more fundamental than Lagrangian mechanics. Our argument provides a non-metaphysical strategy for privileging one formulation of a theory over another: ceteris paribus, a more general formulation is more fundamental. We illustrate this criterion through a novel interpretation of classical mechanics, based on three physical conditions. Two of these conditions suffice for recovering Hamiltonian mechanics. A third condition is necessary for Lagrangian mechanics. Hence, Lagrangian systems are a proper subset of Hamiltonian systems. Finally, we provide a geometric interpretation of the principle of stationary action and rebut arguments for privileging Lagrangian mechanics.