The inverse problem of the calculus of variations for ordinary differential equations

Anderson, Ian; Thompson, Gerard

doi:10.1090/memo/0473

Cited by 92 publications

(174 citation statements)

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“…In particular the multiplier problem for a system of two second-order ordinary differential equations has been thoroughly analyzed in the famous work of J. Douglas [10]. Recently Anderson and Thompson [4] have also studied this problem using the variational bicomplex (see section 2 below). Our solution for the scalar fourth-order equations will be based on ideas from these two articles.…”

Section: E Felsmentioning

confidence: 99%

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The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations

Fels

1996

Trans. Amer. Math. Soc.

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Abstract. A simple invariant characterization of the scalar fourth-order ordinary differential equations which admit a variational multiplier is given. The necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth-order equation through the application of Cartan's equivalence method. The solution to the inverse problem for fourth-order scalar equations provides the solution to an equivalence problem for secondorder Lagrangians, as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated Lagrangian.

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Section: E Felsmentioning

confidence: 99%

“…The approach we take in solving the fourth-order inverse problem follows a refined version of Douglas's solution to the multiplier problem given by Anderson and Thompson [4]. Anderson and Thompson derive the system of determining equations for the multiplier in a natural way using the variational bicomplex.…”

Section: E Felsmentioning

confidence: 99%

The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations

Fels

1996

Trans. Amer. Math. Soc.

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“…An inverse problem [4,5,7,8,9,15] in variational calculus is to find a Lagrangian whose EL equation is that DE. The Lagrangian of a DE is not unique and certain DEs, like scalar-evolution equations, do not admit Lagrangians [5,8].…”

Section: Summary and Discussionmentioning

confidence: 99%

“…The Lagrangian of a DE is not unique and certain DEs, like scalar-evolution equations, do not admit Lagrangians [5,8]. There are several techniques developed to construct Lagrangians for special classes of ODEs, PDEs and their systems by various authors [4,5,7,8,9,15]. The question of finding a Lagrangian is an open problem as most of the approaches apply to some special classes of DEs (both ODEs and PDEs) and their systems.…”

Section: Summary and Discussionmentioning

confidence: 99%

Complex Lie Symmetries for Variational Problems

Ali¹,

Mahomed²,

Qadir³

2008

JNMP

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We present the use of complex Lie symmetries in variational problems by defining a complex Lagrangian and considering its Euler-Lagrange ordinary differential equation. This Lagrangian results in two real "Lagrangians" for the corresponding system of partial differential equations, which satisfy Euler-Lagrange like equations. Those complex Lie symmetries that are also Noether symmetries (i.e. symmetries of the complex Lagrangian) result in two real Noether symmetries of the real "Lagrangians". Also, a complex Noether symmetry of a second order complex ordinary differential equation results in a double reduction of the complex ordinary differential equation. This implies a double reduction in the corresponding system of partial differential equations.

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“…We will call these conditions dynamical consistency conditions. When the noncommutative parameter θ ij is zero, it turns out that these consistency conditions are the Helmholtz conditions for the non restricted inverse problem of the calculus of variations [13] associated with the equations of motion (1) i.e., the integrability conditions of the partial differential equations…”

Section: Consistency Conditions For Ncrmentioning

confidence: 99%

Equations of motion, noncommutativity and quantization

Cortese

Garcı́a

2006

Physics Letters A

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We study the relation between a given set of equations of motion in configuration space and a Poisson bracket. A Poisson structure is consistent with the equations of motion if the symplectic form satisfy some consistency conditions. When the symplectic structure is commutative these conditions are the Helmholtz integrability equations for the nonrestricted inverse problem of the calculus of variations [8]. We have found the corresponding consistency conditions for the symplectic noncommutative case.

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The inverse problem of the calculus of variations for ordinary differential equations

Cited by 92 publications

References 0 publications

The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations

The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations

Complex Lie Symmetries for Variational Problems

Equations of motion, noncommutativity and quantization

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