Criteria for partial differential equations to be Euler-Lagrange equations

This paper presents an investigation on the dynamical properties of single-walled carbon nanotubes (SWCNTs), and nonlinear modal interaction and energy exchange are analysed in detail. Resonance interactions between two conjugate circumferential flexural modes (CFMs) are investigated. The nanotubes are analysed through a continuous shell model, and a thin shell theory is used to model the dynamics of the system; free-free boundary conditions are considered. The Rayleigh–Ritz method is applied to approximate linear eigenfunctions of the partial differential equations that govern the shell dynamics. An energy approach, based on Lagrange equations and series expansion of the displacements, is considered to reduce the initial partial differential equations to a set of nonlinear ordinary differential equations of motion. The model is validated in linear field (natural frequencies) by means of comparisons with literature. A convergence analysis is carried out in order to obtain the smallest modal expansion able to simulate the nonlinear regimes. The time evolution of the nonlinear energy distribution over the SWCNT surface is studied. The nonlinear dynamics of the system is analysed by means of phase portraits. The resonance interaction and energy transfer between the conjugate CFMs are investigated. A travelling wave moving along the circumferential direction of the SWCNT is observed.

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Optimal Feedback Solutions for a Class of Distributed Systems

Journal of Basic Engineering ◽

10.1115/1.3645859 ◽

1966 ◽

Vol 88 (2) ◽

pp. 337-342 ◽

Cited By ~ 11

Author(s):

H. C. Khatri ◽

R. E. Goodson

Keyword(s):

Heat Transfer ◽

Integral Equation ◽

Control System ◽

Partial Differential Equations ◽

Differential Equations ◽

Lagrange Equations ◽

Partial Differential ◽

Approximate Methods ◽

Exact And Approximate Methods ◽

Good Agreement

In the design of controllers for heat transfer systems, one must often describe the plant dynamics by partial differential equations. The problem of optimizing a controller for a system described by partial differential equations is considered here using exact and approximate methods. Results equivalent to the Euler-Lagrange equations are derived for the minimization of an index of performance with integral equation constraints. These integral equation constraints represent the solution of the partial differential equations and the associated boundary conditions. The optimization of the control system using a product expansion as an approximation to the transcendental transfer function of the system is also considered. The results using the two methods are in good agreement. Two examples are given illustrating the application of both the exact and approximate methods. The approximate method requires less computation.

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Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

European Journal of Mathematics ◽

10.1007/s40879-020-00436-7 ◽

2020 ◽

Author(s):

Matteo Petrera ◽

Mats Vermeeren

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Integrable Hierarchies ◽

Dimensional Space ◽

Original System ◽

Lagrange Equations ◽

Partial Differential ◽

Dimensional Space Time ◽

Variational Symmetries ◽

Nonlinear Math

Abstract We investigate the relation between pluri-Lagrangian hierarchies of 2-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in Petrera and Suris (Nonlinear Math. Phys. 24(suppl. 1):121–145, 2017) for ordinary differential equations. We consider hierarchies of 2-dimensional Lagrangian PDEs (many of which have a natural $$(1\,{+}\,1)$$ ( 1 + 1 ) -dimensional space-time interpretation) and show that if the flow of each PDE is a variational symmetry of all others, then there exists a pluri-Lagrangian 2-form for the hierarchy. The corresponding multi-time Euler–Lagrange equations coincide with the original system supplied with commuting evolutionary flows induced by the variational symmetries.

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On a new class of fractional partial differential equations II

Advances in Calculus of Variations ◽

10.1515/acv-2016-0056 ◽

2018 ◽

Vol 11 (3) ◽

pp. 289-307 ◽

Cited By ~ 7

Author(s):

Tien-Tsan Shieh ◽

Daniel E. Spector

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fractional Partial Differential Equations ◽

Lagrange Equations ◽

Partial Differential ◽

Open Problems ◽

New Class ◽

Existence Of Minimizers ◽

Fractional Pde ◽

Regularity Results

AbstractIn this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular, we here establish an {L^{1}} Hardy inequality, obtain further regularity results for solutions of certain fractional PDE, demonstrate the existence of minimizers for integral functionals of the fractional gradient with non-linear dependence in the field, and also establish the existence of solutions to corresponding Euler–Lagrange equations obtained as conditions of minimality. In addition, we pose a number of open problems, the answers to which would fill in some gaps in the theory as well as to establish connections with more classical areas of study, including interpolation and the theory of Dirichlet forms.

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Symmetries and conservation laws of 2-dimensional ideal plasticity

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500006817 ◽

1988 ◽

Vol 31 (3) ◽

pp. 415-439 ◽

Cited By ~ 16

Author(s):

S. I. Senashov ◽

A. M. Vinogradov

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Conservation Laws ◽

Exact Solutions ◽

Continuum Mechanics ◽

Ideal Plasticity ◽

Lagrange Equations ◽

Partial Differential ◽

Higher Symmetries ◽

Symmetry Transformations

Symmetry theory is of fundamental importance in studying systems of partial differential equations. At present algebras of classical infinitesimal symmetry transformations are known for many equations of continuum mechanics [1, 2, 4]. Methods foi finding these algebras go back to S. Lie's works written about 100 years ago. Ir particular, knowledge of symmetry algebras makes it possible to construct effectively wide classes of exact solutions for equations under consideration and via Noether's theorem to find conservation laws for Euler–Lagrange equations. The natural development of Lie's theory is the theory of “higher” symmetries and conservation laws [5].

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