Partial Differential Equations and Group Theory

Pommaret, J.-F.

doi:10.1007/978-94-017-2539-2

Cited by 126 publications

(451 citation statements)

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“…In order to fix the notations, we quote without any proof the "Three Fundamental Theorems of Lie" that will be of constant use in the sequel ( [26]):…”

Section: Definition 22mentioning

confidence: 99%

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics"

Pommaret¹

2012

Continuum Mechanics - Progress in Fundamentals and Engineering Applications

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“…In order to fix the notations, we quote without any proof the "Three Fundamental Theorems of Lie" that will be of constant use in the sequel ( [26]):…”

Section: Definition 22mentioning

confidence: 99%

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics"

Pommaret¹

2012

Continuum Mechanics - Progress in Fundamentals and Engineering Applications

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“…Finally, we recognize in (21) and (24) the components of the co-adjoint map * * * (.) : ad → g g in the dual of left and right invariant basis respectively [24]. Alternatively, applying to each sight of (21)- (22) and (24)- (25) the co-tangent maps…”

Section: Poincaré-chetayev Equations Of a Cosserat Mediummentioning

confidence: 99%

“…1°) It is worth noting that even if the system is continuous and so, in a certain manner infinitedimensional, the group used in the above construction is finite but parameterized by the material manifold D (we say that the group is gauged over D, [24]). This is radically different from fluid mechanics, where the configuration space of an ideal fluid is a group actually infinite dimensional [26].…”

Section: Remarksmentioning

confidence: 99%

“…These equations today known as Poincaré-Chetayev [19][20][21] or EulerPoincaré equations [22] are a generalization of Lagrange equations on a Lie group not forcibly commutative [23,24]. In flexible multibody dynamics, contrary to standard structural dynamics, the elastic bodies of the mechanism endure not only deformations but also rigid overall motions.…”

Section: Introductionmentioning

confidence: 99%

“…The article is structured as follows. After introducing our notations (section 2), we extend the Poincaré-Chetayev equations to the case of a continuum following the construction of Cosserat brothers [24] (section 3). Then, doing several remarks (section 4), we finally apply (section 5) these equations to a relevant case of flexible multi-body sytem: an open loop flexible manipulator.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Poincaré-Chetayev equations and flexible multibody systems

Boyer¹,

Primault²

2005

Journal of Applied Mathematics and Mechanics

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This article is devoted to the dynamics of flexible multi-body systems and to their links with a fundamental set of equations discovered by H. Poincaré one hundred years ago [1]. These equations, called "Poincaré-Chetayev equations", are today known to be the foundation of the Lagrangian reduction theory. Starting with the extension of these equations to a Cosserat medium, we show that the two basic sets of equations used in flexible multi-body dynamics. The generalized Newton-Euler model of flexible multi-body systems in the floating frame approach and the partial differential equations of the nonlinear geometrically exact theory in the Galilean approach, are Poincaré-Chetayev equations.

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5 Algebraic Analysis of Control Systems Defined by Partial Differential Equations

Pommaret

2005

Advanced Topics in Control Systems Theory

129

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The present chapter contains the material taught within the module P2 of FAP 2004. The purpose of this intensive course is first to provide an introduction to "algebraic analysis". This fashionable though quite difficult domain of pure mathematics today has been pioneered by V.P. Palamodov, M. Kashiwara and B. Malgrange around 1970, after the work of D.C. Spencer on the formal theory of systems of partial differential equations. We shall then focus on its application to control theory in order to study linear control systems defined by partial differential equations with constant or variable coefficients, also called multidimensional control systems, by means of new methods from module theory and homological algebra. We shall revisit a few basic concepts and prove, in particular, that controllability, contrary to a well established engineering tradition or intuition, is an intrinsic structural property of a control system, not depending on the choice of inputs and outputs among the control variables or even on the presentation of the control system. Our exposition will be rather elementary as we shall insist on the main ideas and methods while illustrating them through explicit examples. Meanwhile, we want to stress out the fact that these new techniques bring striking results even on classical control systems of Kalman type !.

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Partial Differential Equations and Group Theory

Cited by 126 publications

References 0 publications

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics"

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics"

The Poincaré-Chetayev equations and flexible multibody systems

5 Algebraic Analysis of Control Systems Defined by Partial Differential Equations

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