Contact Geometry and Non-linear Differential Equations

Kushner, Alexei G.; Lychagin, Valentin; Rubtsov, Vladimir

doi:10.1017/cbo9780511735141

Cited by 155 publications

(241 citation statements)

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“…The Lagrangian and the Hamiltonian formalisms have geometric interpretation based on symplectic geometry (see, for example, [5] for a treatise on symplectic geometry and [6] for some modern applications). Nowadays geometric approach in mechanics is under active study [7,8].…”

Section: )mentioning

confidence: 99%

Geometric Hamiltonian formalism for reparameterization-invariant theories with higher derivatives

Dunin-Barkowski

Слепцов

2009

Theor Math Phys

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Reparametrization invariant Lagrangian theories with higher derivatives are considered. We investigate the geometric structures behind these theories and construct the Hamiltonian formalism in a geometric way. The Legendre transformation which corresponds to the transition from the Lagrangian formalism to the Hamiltonian formalism is non-trivial in this case. The resulting phase bundle, i.e. the image of the Legendre transformation, is a submanifold of some cotangent bundle. We show that in our construction it is always odd-dimensional. Therefore the canonical symplectic two-form from the ambient cotangent bundle generates on the phase bundle a field of the null-directions of its restriction. It is shown that the integral lines of this field project directly to the extremals of the action on the configuration manifold. Therefore this naturally arising field is what is called the Hamilton field. We also express the corresponding Hamilton equations through the generilized Nambu bracket.

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Section: )mentioning

confidence: 99%

Geometric Hamiltonian formalism for reparameterization-invariant theories with higher derivatives

Dunin-Barkowski

Слепцов

2009

Theor Math Phys

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“…In particular, this was done in a systematic manner in the dissertation of A. Kushner who synthesized previously proposed approaches and techniques. His results are reported in book [6]. Kushner's approach is based on the use of machinery of effective forms in contact geometry, proposed to this end by V. Lychagin at late 70s and since that time actively exploited by himself and his collaborators.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, a hyperbolic MAE of this kind is completely characterized by an associated 2-dimensional non-Lagrangian distribution D on a symplectic 4-fold M and its solutions are interpreted as Lagrangian submanifolds L of M such that the restriction of D to L is one-dimensional. By following the terminology of [6] we call symplectic MAEs of this kind (shortly, SMAEs).…”

Section: Introductionmentioning

confidence: 99%

“…For instance, in book [6] just few of them are mentioned. On the other hand, SDIs are of crucial importance for the equivalence and classification problems, for practical computations of symmetries and conservation laws, etc.…”

Section: Introductionmentioning

confidence: 99%

“…The reader can find an extensive bibliography dedicated to history of topics considered in our paper in book [6].…”

Section: Introductionmentioning

confidence: 99%

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Scalar differential invariants of symplectic Monge-Ampère equations

Paris

Vinogradov

2011

Open Mathematics

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Abstract. All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allows us to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Ampère equations. As an example we study equations of the form uxy + f (x, y, ux, uy) = 0 and in particular find a simple linearization criterion.

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From modelling of systems with constraints to generalized geometry and back to numerics

Salnikov

Hamdouni

2019

Z Angew Math Mech

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In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints -we recall the mathematical construction in the context of so called implicit Lagrangian systems. We explain how they can be used to produce new numerical methods, that we call Dirac integrators.On a test example of a simple pendulum in a gravity field we compare the Dirac integrators with classical explicit and implicit methods, we pay special attention to conservation of constrains. Then, on a more advanced example of the Ziegler-type system we show that the choice of numerical methods can indeed affect the conclusions of qualitative analysis of the dynamics of mechanical systems. We also tell why we think that Dirac integrators are appropriate for this kind of systems by explaining the relation with the notions of geometric degree of non-conservativity and kinematic structural stability. K E Y W O R D SDirac structures, geometric integrators, systems with constraints

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Contact Geometry and Non-linear Differential Equations

Cited by 155 publications

References 0 publications

Geometric Hamiltonian formalism for reparameterization-invariant theories with higher derivatives

Geometric Hamiltonian formalism for reparameterization-invariant theories with higher derivatives

Scalar differential invariants of symplectic Monge-Ampère equations

From modelling of systems with constraints to generalized geometry and back to numerics

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