A Variational Principle for Non‐Conservative Dynamical Systems

Vujanović, B.

doi:10.1002/zamm.19750550605

Cited by 23 publications

(7 citation statements)

References 3 publications

(3 reference statements)

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“…A conservative system has f = 0, which yields the familiar Euler-Lagrange equation. Moreover, according to [19], the principle (10) relates to the d'Alembert principle of virtual work.…”

Section: B Hamilton and Extended Hamilton Principle: Law Of Motion Amentioning

confidence: 99%

Extended Hamiltonian Learning on Riemannian Manifolds: Theoretical Aspects

Fiori

2011

IEEE Trans. Neural Netw.

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This paper introduces a general theory of extended Hamiltonian (second-order) learning on Riemannian manifolds, as an instance of learning by constrained criterion optimization. The dynamical learning equations are derived within the general framework of extended-Hamiltonian stationary-action principle and are expressed in a coordinate-free fashion. A theoretical analysis is carried out in order to compare the features of the dynamical learning theory with the features exhibited by the gradient-based ones. In particular, gradient-based learning is shown to be an instance of dynamical learning, and the classical gradient-based learning modified by a "momentum" term is shown to resemble discrete-time dynamical learning. Moreover, the convergence features of gradient-based and dynamical learning are compared on a theoretical basis. This paper discusses cases of learning by dynamical systems on manifolds of interest in the scientific literature, namely, the Stiefel manifold, the special orthogonal group, the Grassmann manifold, the group of symmetric positive definite matrices, the generalized flag manifold, and the real symplectic group of matrices.

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“…A conservative system has f = 0, which yields the familiar Euler-Lagrange equation. Moreover, according to [19], the principle (10) relates to the d'Alembert principle of virtual work.…”

Section: B Hamilton and Extended Hamilton Principle: Law Of Motion Amentioning

confidence: 99%

Extended Hamiltonian Learning on Riemannian Manifolds: Theoretical Aspects

Fiori

2011

IEEE Trans. Neural Netw.

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“…In classical mechanics, invariance of action is postulated by using a L function. The method is useful, yet the motivation is very formal [22]. L function leads to E -L equations that are the field equations for the bulk.…”

Section: Introductionmentioning

confidence: 99%

Energy based methods applied in mechanics by using the extended Noether's formalism

Abali¹

2022

Preprint

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Physical systems are modeled by field equations, these are coupled, partial differential equations in space and time. Field equations are often given by balance equations and constitutive equations, where the former is axiomatically given and the latter is thermodynamically derived. This method has difficulties in damage mechanics in generalized continua. An alternative method to derive governing equations is based on a variational framework known as the extended N 's formalism. Albeit being very prominent, its use is less known in applied mechanics as a field theory. In this work, we demonstrate the power of extended N 's formalism by using tensor algebra and usual continuum mechanics nomenclature.

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“…Hamilton's principle is extended to the holonomic nonconservative system [4], the high-order system [5], and the nonholonomic system [4]. In addition, the Pfaff-Birkhoff principle [6,7], generalized Pfaff-Birkhoff principle [8], and Vujanović's variational principle of nonconservative system [9] are also the generalization of Hamilton's principle.…”

Section: Introductionmentioning

confidence: 99%

Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

Song

Yao

2021

Advances in Mathematical Physics

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Based on the generalized operators, Hamilton equation, Noether symmetry, and perturbation to Noether symmetry are studied. The main contents are divided into four parts, and every part includes two generalized operators. Firstly, Hamilton equations within generalized operators are established. Secondly, the Noether symmetry method and conserved quantity are studied. Thirdly, perturbation to the Noether symmetry and adiabatic invariant are presented. And finally, two applications are presented to illustrate the methods and results.

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A Variational Principle for Non‐Conservative Dynamical Systems

Cited by 23 publications

References 3 publications

Extended Hamiltonian Learning on Riemannian Manifolds: Theoretical Aspects

Extended Hamiltonian Learning on Riemannian Manifolds: Theoretical Aspects

Energy based methods applied in mechanics by using the extended Noether's formalism

Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

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