Darboux Theorem Research Articles

Darboux, Moser and Weinstein theorems for prequantum systems and applications to geometric quantization

We establish analogs of the Darboux, Moser and Weinstein theorems for prequantum systems. We show that two prequantum systems on a manifold with vanishing first cohomology, with symplectic forms defining the same cohomology class and homotopic to each other within that class, differ only by a symplectomorphism and a gauge transformation. As an application, we show that the Bohr-Sommerfeld quantization of a prequantum system on a manifold with trivial first cohomology is independent of the choice of the connection.

On Darboux theorems for geometric structures induced by closed forms

This work reviews the classical Darboux theorem for symplectic, presymplectic, and cosymplectic manifolds (which are used to describe mechanical systems), as well as certain cases of multisymplectic manifolds, while extends the Darboux theorem in new ways to k-symplectic and k-cosymplectic manifolds (all these structures appear in the geometric formulation of first-order classical field theories). Moreover, we discuss the existence of Darboux theorems for classes of precosymplectic, k-presymplectic, k-precosymplectic, and premultisymplectic manifolds, which are the geometrical structures underlying some kinds of singular field theories, i.e. with locally non-invertible Legendre maps. Approaches to Darboux theorems based on flat connections associated with geometric structures are given, while new results on polarisations for (k-)(pre)(co)symplectic structures arise.

The coupled Boussinesq equation and its Darboux transformation on the time–space scale

Gel’fand-Dikii (GD) formalism is an approach for generating integrable systems in terms of fractional powers of the δ differential operator. In this paper, it extends the GD formalism associated with the third-order δ differential operator L to the time scale. Then, the coupled Boussinesq equation on the time–space scale is given by taking special values, and it can be reduced on different time–space scales. Moreover, the exact solutions of the coupled Boussinesq equation on the time–space scale and the classical Boussinesq equation are constructed via employing the extensions of the Darboux theorem and Crum theorem on the time scale.

Remarks on the infinite‐dimensional counterparts of the Darboux theorem

The Darboux theorem, one of the fundamental results in analysis, states that the derivative of a real (not necessarily continuously) differentiable function defined on a compact interval has the intermediate value property, that is, attains each value between the derivatives at the endpoints. The Bolzano intermediate value theorem, which implies Darboux's theorem when the derivative is continuous, states that a continuous real‐valued function defined on satisfying and , has a zero, that is, for at least one number . It has numerous counterparts in multivariate calculus as well as in the infinite‐dimensional setting. The present paper is devoted to the discussion of some infinite‐dimensional variants of the Darboux theorem, which does not seem to be sufficiently deeply discussed. The study relies on different notions of nonsmooth differentiability of real functions and some appropriate compactness conditions. Problems involving functionals bounded below, monotone operators, and some general questions concerning the existence of the so‐called generalized equilibria are discussed.

Lipschitz Homotopy Groups of Contact 3-Manifolds

We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group $\mathbb H^n$ with its Carnot-Carathéodory metric. Then each contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is purely $k$-unrectifiable for $k>n$. We then extend results of Dejarnette et al. [5] and Wenger and Young [20] on the Lipschitz homotopy groups of $\mathbb H^1$ to an arbitrary contact 3-manifold endowed with a Carnot-Carathéodory metric, namely that for any contact 3-manifold the first Lipschitz homotopy group is uncountably generated and all higher Lipschitz homotopy groups are trivial. Therefore, in the sense of Lipschitz homotopy groups, a contact 3-manifold is a $K(\pi,1)$-space with an uncountably generated first homotopy group. Along the way, we prove that each open distributional embedding between purely 2-unrectifiable sub-Riemannian manifolds induces an injective map on the associated first Lipschitz homotopy groups. Therefore, each open subset of a contact 3-manifold determines an uncountable subgroup of the first Lipschitz homotopy group of the contact 3-manifold.

Quantum Darboux theorem

The problem of computing quantum mechanical propagators can be recast as a computation of a Wilson line operator for parallel transport by a flat connection acting on a vector bundle of wavefunctions. In this picture the base manifold is an odd dimensional symplectic geometry, or quite generically a contact manifold that can be viewed as a "phase-spacetime", while the fibers are Hilbert spaces. This approach enjoys a "quantum Darboux theorem" that parallels the Darboux theorem on contact manifolds which turns local classical dynamics into straight lines. We detail how the quantum Darboux theorem works for anharmonic quantum potentials. In particular, we develop a novel diagrammatic approach for computing the asymptotics of a gauge transformation that locally makes complicated quantum dynamics trivial.

Generalization of Hamiltonian mechanics to a three-dimensional phase space

Abstract Classical Hamiltonian mechanics is realized by the action of a Poisson bracket on a Hamiltonian function. The Hamiltonian function is a constant of motion (the energy) of the system. The properties of the Poisson bracket are encapsulated in the symplectic $2$-form, a closed second-order differential form. Due to closure, the symplectic $2$-form is preserved by the Hamiltonian flow, and it assigns an invariant (Liouville) measure on the phase space through the Lie–Darboux theorem. In this paper we propose a generalization of classical Hamiltonian mechanics to a three-dimensional phase space: the classical Poisson bracket is replaced with a generalized Poisson bracket acting on a pair of Hamiltonian functions, while the symplectic $2$-form is replaced by a symplectic $3$-form. We show that, using the closure of the symplectic $3$-form, a result analogous to the classical Lie–Darboux theorem holds: locally, there exist smooth coordinates such that the components of the symplectic $3$-form are constants, and the phase space is endowed with a preserved volume element. Furthermore, as in the classical theory, the Jacobi identity for the generalized Poisson bracket mathematically expresses the closure of the associated symplectic form. As a consequence, constant skew-symmetric third-order contravariant tensors always define generalized Poisson brackets. This is in contrast with generalizations of Hamiltonian mechanics postulating the fundamental identity as replacement for the Jacobi identity. In particular, we find that the fundamental identity represents a stronger requirement than the closure of the symplectic $3$-form.

Projective limit of a sequence of compatible weak symplectic forms on a sequence of Banach bundles and Darboux Theorem

Given a projective sequence of Banach bundles, each one provided with of a weak symplectic form, we look for conditions under which, the corresponding sequence of weak symplectic forms gives rise to weak symplectic form on the projective limit bundle. Then we apply this results to the tangent bundle of a projective limit of Banach manifolds. This naturally leads to ask about conditions under which the Darboux Theorem is also true on the projective limit of Banach manifolds. We will give some necessary and some sufficient conditions so that such a result is true. Then we discuss why, in general, the Moser's method can not work on projective limit of Banach weak symplectic Banach manifolds without very strong conditions like Kumar's results ([15]). In particular we give an analog result of Kumar' s one with weaker assumptions and we give an example for which such weaker conditions are satisfied. More generally, we produce examples of projective sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true and an example for which the Darboux Theorem is true on each manifold, but is not true on the projective limit of these manifolds.

A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Pantev, Toen, Vaqui\'e and Vezzosi arXiv:1111.3209 defined $k$-shifted symplectic derived schemes and stacks ${\bf X}$ for $k\in\mathbb Z$, and Lagrangians ${\bf f}:{\bf L}\to{\bf X}$ in them. They have important applications to Calabi-Yau geometry and quantization. Bussi, Brav and Joyce arXiv:1305.6302 proved a 'Darboux Theorem' giving explicit Zariski or \'etale local models for $k$-shifted symplectic derived schemes ${\bf X}$ for $k<0$ presenting them as twisted shifted cotangent bundles. We prove a 'Lagrangian Neighbourhood Theorem' giving explicit Zariski or etale local models for Lagrangians ${\bf f}:{\bf L}\to{\bf X}$ in $k$-shifted symplectic derived schemes ${\bf X}$ for $k<0$, relative to the Bussi-Brav-Joyce 'Darboux form' local models for ${\bf X}$. That is, locally such Lagrangians can be presented as twisted shifted conormal bundles. We also give a partial result when $k=0$. We expect our results will have future applications to $k$-shifted Poisson geometry (see arXiv:1506.03699), to defining 'Fukaya categories' of complex or algebraic symplectic manifolds, and to categorifying Donaldson-Thomas theory of Calabi-Yau 3-folds and 'Cohomological Hall algebras'.

On the generalization of the Darboux theorem

Darboux theorem to more general context of Frechet manifolds we face an obstacle: in general vector fields do not have local flows. Recently, Fr\'{e}chet geometry has been developed in terms of projective limit of Banach manifolds. In this framework under an appropriate Lipchitz condition The Darboux theorem asserts that a symplectic manifold $(M^{2n},\omega)$ is locally symplectomorphic to $(R^{2n}, \omega_0)$, where $\omega_0$ is the standard symplectic form on $R^{2n}$. This theorem was proved by Moser in 1965, the idea of proof, known as the Moser’s trick, works in many situations. The Moser tricks is to construct an appropriate isotopy $ \ff_t $ generated by a time-dependent vector field $ X_t $ on $M$ such that $ \ff_1^{*} \omega = \omega_0$. Nevertheless, it was showed by Marsden that Darboux theorem is not valid for weak symplectic Banach manifolds. However, in 1999 Bambusi showed that if we associate to each point of a Banach manifold a suitable Banach space (classifying space) via a given symplectic form then the Moser trick can be applied to obtain the theorem if the classifying space does not depend on the point of the manifold and a suitable smoothness condition holds.&#x0D; If we want to try to generalize the local flows exist and with some restrictive conditions the Darboux theorem was proved by Kumar. In this paper we consider the category of so-called bounded Fr\'{e}chet manifolds and prove that in this category vector fields have local flows and following the idea of Bambusi we associate to each point of a manifold a Fr\'{e}chet space independent of the choice of the point and with the assumption of bounded smoothness on vector fields we prove the Darboux theorem.

A note on some recent results of the conformable fractional derivative

In this note, we discuss, improve and complement some recent results of the conformable fractional derivative introduced and established by Katugampola[arxiv:1410.6535v1] and Khalil et al. [J. Comput. Appl. Math. 264(2014) 65-70]. Among other things we show that each function $f$ defined on $(a,b)$, $a&amp;gt;0$ has a conformable fractional derivative (CFD) if and only if it has a classical first derivative. At the end of the paper, we prove the Rolle's, Cauchy, Lagrange's and Darboux's theorem in the context of Conformable Fractional Derivatives.

Local representation and construction of Beltrami fields

A Beltrami field is an eigenvector of the curl operator. Beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence. By application of the Lie–Darboux theorem of differential geometry, we prove a local representation theorem for Beltrami fields. We find that, locally, a Beltrami field has a standard form amenable to an Arnold–Beltrami–Childress flow with two of the parameters set to zero. Furthermore, a Beltrami flow admits two local invariants, a coordinate representing the physical plane of the flow, and an angular momentum-like quantity in the direction across the plane. As a consequence of the theorem, we derive a method to construct Beltrami fields with given proportionality factor. This method, based on the solution of the eikonal equation, guarantees the existence of Beltrami fields for any orthogonal coordinate system such that at least two scale factors are equal. We construct several solenoidal and non-solenoidal Beltrami fields with both homogeneous and inhomogeneous proportionality factors.

On Darboux Theorem for symplectic forms on direct limits of symplectic Banach manifolds

Given an ascending sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true, we can ask about conditions under which the Darboux Theorem is also true on the direct limit. We will show that, in general, without very strong conditions, the answer is negative. In particular, we give an example of an ascending symplectic Banach manifolds on which the Darboux Theorem is true but not on the direct limit. In the second part, we illustrate this discussion in the context of an ascending sequence of Sobolev manifolds of loops in symplectic finite-dimensional manifolds. This context gives rise to an example of direct limit of weak symplectic Banach manifolds on which the Darboux Theorem is true around any point.

A Generalization of an Integrability Theorem of Darboux

In his monograph “Systemes Orthogonaux” (Darboux, in Lecons sur les systemes orthogonaux et les coordonnees curvilignes, Gauthier-Villars, Paris, 1910) Darboux stated three theorems providing local existence and uniqueness of solutions to first-order systems of the type $$\begin{aligned} \partial _{x_i} u_\alpha (x)=f^\alpha _i(x,u(x)), \quad i\in I_\alpha \subseteq \{1,\dots ,n\}. \end{aligned}$$For a given point $${\bar{x}}\in \mathbb {R}^n$$ it is assumed that the values of the unknown $$u_\alpha $$ are given locally near $${\bar{x}}$$ along $$\{x\,|\, x_i={\bar{x}}_i \, \text {for each}\, i\in I_\alpha \}$$. The more general of the theorems, Theoreme III, was proved by Darboux only for the cases $$n=2$$ and 3. In this work we formulate and prove a generalization of Darboux’s Theoreme III which applies to systems of the form $$\begin{aligned} {{\mathbf {r}}}_i(u_\alpha )\big |_x = f_i^\alpha (x, u(x)), \quad i\in I_\alpha \subseteq \{1,\dots ,n\} \end{aligned}$$where $${\mathcal {R}}=\{{{\mathbf {r}}}_i\}_{i=1}^n$$ is a fixed local frame of vector fields near $${\bar{x}}$$. The data for $$u_\alpha $$ are prescribed along a manifold $$\Xi _\alpha $$ containing $${\bar{x}}$$ and transverse to the vector fields $$\{{{\mathbf {r}}}_i\,|\, i\in I_\alpha \}$$. We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the frame $${\mathcal {R}}$$ and on the manifolds $$\Xi _\alpha $$; it is automatically met in the case considered by Darboux. Assuming the SCC and the relevant integrability conditions are satisfied, we establish local existence and uniqueness of a $$C^1$$-solution via Picard iteration for any number of independent variables n.

A Darboux theorem for derived schemes with shifted symplectic structure

We prove a Darboux theorem for derived schemes with symplectic forms of degree k &gt; 0 k&gt;0 , in the sense of Pantev, Toën, Vaquié, and Vezzosi. More precisely, we show that a derived scheme X \mathbfit {X} with symplectic form ω ~ \tilde {\omega } of degree k k is locally equivalent to ( Spec ⁡ A , ω ) (\operatorname {Spec} A,\omega ) for Spec ⁡ A \operatorname {Spec} A an affine derived scheme in which the cdga A A has Darboux-like coordinates with respect to which the symplectic form ω \omega is standard, and in which the differential in A A is given by a Poisson bracket with a Hamiltonian function Φ \Phi of degree k + 1 k+1 . When k = − 1 k=-1 , this implies that a − 1 -1 -shifted symplectic derived scheme ( X , ω ~ ) (\mathbfit {X}, \tilde {\omega }) is Zariski locally equivalent to the derived critical locus Crit ⁡ ( Φ ) \operatorname {Crit}(\Phi ) of a regular function Φ : U → A 1 \Phi :U\rightarrow \mathbb {A}^1 on a smooth scheme U U . We use this to show that the classical scheme X = t 0 ( X ) X=t_0(\mathbfit {X}) has the structure of an algebraic d-critical locus, in the sense of Joyce. In a series of works, the authors and their collaborators extend these results to (derived) Artin stacks, and discuss a Lagrangian neighbourhood theorem for shifted symplectic derived schemes, and applications to categorified and motivic Donaldson–Thomas theory of Calabi–Yau 3-folds, and to defining new Donaldson–Thomas type invariants of Calabi–Yau 4-folds, and to defining Fukaya categories of Lagrangians in algebraic symplectic manifolds using perverse sheaves.

Symplectic factorization, Darboux theorem and ellipticity

This manuscript identifies a maximal system of equations which renders the classical Darboux problem elliptic, thereby providing a selection criterion for its well posedness. Let f be a symplectic form close enough to ωm, the standard symplectic form on R2m. We prove existence of a diffeomorphism φ, with optimal regularity, satisfyingφ⁎(ωm)=fand〈dφ♭;ωm〉=0. We establish uniqueness of φ when the system is coupled with a Dirichlet datum. As a byproduct, we obtain, what we term symplectic factorization of vector fields, that any map u, satisfying appropriate assumptions, can be factored as:u=χ∘ψwithψ⁎(ωm)=ωm,〈dχ♭;ωm〉=0and∇χ+(∇χ)t>0; moreover there exists a closed 2-form Φ such that χ=(δΦ⌟ωm)♯. Here, ♯ is the musical isomorphism and ♭ its inverse. We connect the above result to an L2-projection problem.

On Finding Geodesic Equation of Student T Distribution

Student t distribution has been widely applied in the course of statistics. In this paper, we focus on finding a geodesic equation of the two parameter student t distributions. To find this equation, we applied both the well-known Darboux Theorem and a triply of partial differential equations taken from Struik D.J. (Struik, D.J., 1961) or Grey A (Grey A., 1993), As expected, the two different approaches reach the same type of results. The solution proposed in this paper could be used as a general solution of the geodesic equation for the student t distribution.

Lagrange’s theorem, convex functions and Gauss map

As one of the main results we prove that if f has Lagrange unique property then f is strictly convex or concave (we do not assume continuity of the derivative), Theorem 2.1. We give two different proofs of Theorem 2.1 (one mainly using Lagrange theorem and the other using Darboux theorem). In addition, we give a few characterizations of strictly convex curves, in Theorem 3.5. As an application of it, we give characterization of strictly convex planar curves, which have only tangents at every point, by injective of the Gauss map. Also without the differentiability hypothesis we get the characterization of strictly convex or concave functions by two points property, Theorem 4.2.

Quantitative Darboux theorems in contact geometry

This paper begins the study of relations between Riemannian geometry and contact topology in any dimension and continues this study in dimension 3. Specifically we provide a lower bound for the radius of a geodesic ball in a contact manifold that can be embedded in the standard contact structure on Euclidean space, that is on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form. In dimension three, it further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are classical comparison theorems in Riemannian geometry. In the same context, we also use holomorphic curves techniques to provide a lower bound for the radius of a PS-tight ball.

A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications

This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This paper extends the previous three from (derived) schemes to (derived) Artin stacks. We prove four main results: (a) If $(X,\omega)$ is a $k$-shifted symplectic derived Artin stack for $k<0$ in the sense of arXiv:1111.3209, then near each $x\in X$ we can find a 'minimal' smooth atlas $\varphi:U\to X$ with $U$ an affine derived scheme, such that $(U,\varphi^*(\omega))$ may be written explicitly in coordinates in a standard 'Darboux form'. (b) If $(X,\omega)$ is a $-1$-shifted symplectic derived Artin stack and $X'$ the underlying classical Artin stack, then $X'$ extends naturally to a 'd-critical stack' $(X',s)$ in the sense of arXiv:1304.4508. (c) If $(X,s)$ is an oriented d-critical stack, we can define a natural perverse sheaf $P^\bullet_{X,s}$ on $X$, such that whenever $T$ is a scheme and $t:T\to X$ is smooth of relative dimension $n$, then $T$ is locally modelled on a critical locus Crit$(f:U\to{\mathbb A}^1)$ for $U$ smooth, and $t^*(P^\bullet_{X,s})[n]$ is locally modelled on the perverse sheaf of vanishing cycles $PV_{U,f}^\bullet$ of $f$. (d) If $(X,s)$ is a finite type oriented d-critical stack, we can define a natural motive $MF_{X,s}$ in a ring of motives $\bar{\mathcal M}^{st,\hat\mu}_X$ on $X$, such that whenever $T$ is a finite type scheme and $t:T\to X$ is smooth of dimension $n$, then $T$ is locally modelled on a critical locus Crit$(f:U\to{\mathbb A}^1)$ for $U$ smooth, and ${\mathbb L}^{-n/2}\odot t^*(MF_{X,s})$ is locally modelled on the motivic vanishing cycle $MF^{mot,\phi}_{U,f}$ of $f$ in $\bar{\mathcal M}^{st,\hat\mu}_T$. Our results have applications to categorified and motivic extensions of Donaldson-Thomas theory of Calabi-Yau 3-folds