Méchanique analitique

Lagrange, J. L; Kelvin, William Thomson; Howie, William

doi:10.5479/sil.322586.39088000898585

Cited by 173 publications

(19 citation statements)

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“…The classical mechanics was reformulated as Lagrangian mechanics by Joseph Louis Lagrange in 1788 [2]. In 1833, William Rowan Hamilton [3] formulated the Hamiltonian mechanics by utilizing the Legendre transformation [4,5].…”

Section: Introductionmentioning

confidence: 99%

“…There are three major approaches to deal with dynamic optimization problems: calculus of variations, dynamic programming and optimal control theory. The calculus of variation utilizes the notion of a standard Lagrangian and provides a set of equations known as Euler-Lagrange equations [2,6]. The dynamic programming was introduced by Richard Ernest Bellman [7].…”

Section: Introductionmentioning

confidence: 99%

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The applications of the partial Hamiltonian approach to mechanics and other areas

Naz

2016

International Journal of Non-Linear Mechanics

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The partial Hamiltonian systems of the formq i = ∂H ∂p i ,ṗ i = − ∂H ∂q i + Γ i (t, q i , p i ) arise widely in different fields of the applied mathematics. The partial Hamiltonian systems appear for a mechanical system with non-holonomic nonlinear constraints and non-potential generalized forces. In dynamic optimization problems of economic growth theory involving a non-zero discount factor the partial Hamiltonian systems arise and are known as a current value Hamiltonian systems. It is shown that the partial Hamiltonian approach proposed earlier for the current value Hamiltonian systems arising in economic growth theory Naz et al [1] is applicable to mechanics and other areas as well. The partial Hamiltonian approach is utilized to construct first integrals and closed form solutions of optimal growth model with environmental asset, equations of motion for a mechanical system with non-potential forces, the force-free Duffing Van Der Pol Oscillator and Lotka-Volterra models.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The applications of the partial Hamiltonian approach to mechanics and other areas

Naz

2016

International Journal of Non-Linear Mechanics

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“…It is actually there, but only in an indirect way: Lagrange points out that for a rather general class of mechanical systems, to which the principle of live forces is applicable, a least action formulation leads to the same equations as his virtual velocity formulation. 43 Here the principle of live forces is essentially equivalent to the modern statement of total energy conservation.…”

Section: Discussionmentioning

confidence: 99%

A contemporary look at Hermann Hankel’s 1861 pioneering work on Lagrangian fluid dynamics

Frisch

Grimberg²,

Villone

2017

EPJ H

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The present paper is a companion to the paper by Villone and Rampf (2017), titled "Hermann Hankel's On the general theory of motion of fluids, an essay including an English translation of the complete Preisschrift from 1861" together with connected documents. Here we give a critical assessment of Hankel's work, which covers many important aspects of fluid dynamics considered from a Lagrangian-coordinates point of view: variational formulation in the spirit of Hamilton for elastic (barotropic) fluids, transport (we would now say Lie transport) of vorticity, the Lagrangian significance of Clebsch variables, etc. Hankel's work is also put in the perspective of previous and future work. Hence, the action spans about two centuries: from Lagrange's 1760-1761 Turin paper on variational approaches to mechanics and fluid mechanics problems to Arnold's 1966 founding paper on the geometrical/variational formulation of incompressible flow. The 22-year old Hankel -who was to die 12 years later -emerges as a highly innovative master of mathematical fluid dynamics, fully deserving Riemann's assessment that his Preisschrift contains "all manner of good things."

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“…. , x n ) [16,17]. This smooth function gives the position x of a particular fluid particle given by "label vector" a = (a 1 , .…”

Section: Lagrangian Coordinatesmentioning

confidence: 99%

Symmetries and conservation laws of the Euler equations in Lagrangian coordinates

Shankar¹

2017

Journal of Mathematical Analysis and Applications

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We consider the Euler equations of incompressible inviscid fluid dynamics. We discuss a variational formulation of the governing equations in Lagrangian coordinates. We compute variational symmetries of the action functional and generate corresponding conservation laws in Lagrangian coordinates. We clarify and demonstrate relationships between symmetries and the classical balance laws of energy, linear momentum, center of mass, angular momentum, and the statement of vorticity advection. Using a newly obtained scaling symmetry, we obtain a new conservation law for the Euler equations in Lagrangian coordinates in n-dimensional space. The resulting integral balance relates the total kinetic energy to a new integral quantity defined in Lagrangian coordinates. This relationship implies an inequality which describes the radial deformation of the fluid, and shows the non-existence of time-periodic solutions with nonzero, finite energy.

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Méchanique analitique

Cited by 173 publications

References 0 publications

The applications of the partial Hamiltonian approach to mechanics and other areas

The applications of the partial Hamiltonian approach to mechanics and other areas

A contemporary look at Hermann Hankel’s 1861 pioneering work on Lagrangian fluid dynamics

Symmetries and conservation laws of the Euler equations in Lagrangian coordinates

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