Lagrangian Structures Research Articles

A kind of Lagrangian chaotic property of the Arnold–Beltrami–Childress flow

Three-dimensional steady-state Arnold–Beltrami–Childress (ABC) flow has a chaotic Lagrangian structure, and also satisfies the Navier–Stokes (NS) equations with an external force per unit mass. It is well known that, although trajectories of a chaotic system have sensitive dependence on initial conditions, i.e. the famous ‘butterfly effect’, their statistical properties are often insensitive to small disturbances. This kind of chaos (such as governed by the Lorenz equations) is called normal-chaos. However, a new concept, i.e. ultra-chaos, has been reported recently, whose statistics are unstable to tiny disturbances. Thus, ultra-chaos represents higher disorder than normal chaos. In this paper, we illustrate that ultra-chaos widely exists in Lagrangian trajectories of fluid particles in steady-state ABC flow. Moreover, solving the NS equation when $Re=50$ with the ABC flow plus a very small disturbance as the initial condition, it is found that trajectories of nearly all fluid particles become ultra-chaotic when the transition from laminar to turbulence occurs. These numerical experiments and facts highly suggest that ultra-chaos should have a relationship with turbulence. This paper identifies differences between ultra-chaos and sensitivity of statistics to parameters. Possible relationships between ultra-chaos and the Poincaré section, ultra-chaos and ergodicity/non-ergodicity, etc., are discussed. The concept of ultra-chaos opens a new perspective of chaos, the Poincaré section, ergodicity/non-ergodicity, turbulence and their inter-relationships.

Lagrangian multiforms on Lie groups and non-commuting flows

We describe a variational framework for non-commuting flows, extending the theories of Lagrangian multiforms and pluri-Lagrangian systems, which have gained prominence in recent years as a variational description of integrable systems in the sense of multidimensional consistency. In the context of non-commuting flows, the manifold of independent variables, often called multi-time, is a Lie group whose bracket structure corresponds to the commutation relations between the vector fields generating the flows. Natural examples are provided by superintegrable systems for the case of Lagrangian 1-form structures, and integrable hierarchies on loop groups in the case of Lagrangian 2-forms. As particular examples we discuss the Kepler problem, the rational Calogero-Moser system, and a generalisation of the Ablowitz-Kaup-Newell-Segur system with non-commuting flows. We view this endeavour as a first step towards a purely variational approach to Lie group actions on manifolds.

Lagrangian 3-form structure for the Darboux system and the KP hierarchy

A Lagrangian multiform structure is established for a generalisation of the Darboux system describing orthogonal curvilinear coordinate systems. It has been shown in the past that this system of coupled PDEs is in fact an encoding of the entire Kadomtsev–Petviashvili (KP) hierarchy in terms so-called Miwa variables. Thus, in providing a Lagrangian description of this multidimensionally consistent system amounts to a new Lagrangian 3-form structure for the continuous KP system. A generalisation to the matrix (also known as non-Abelian) KP system is discussed.

Neural vortex method: From finite Lagrangian particles to infinite dimensional Eulerian dynamics

In fluid analysis, there has been a long-standing problem: lacking a rigorous mathematical tool to map from a continuous flow field to finite discrete particles, hurdling the Lagrangian particles from inheriting the high resolution of a large-scale Eulerian solver. To tackle this challenge, we propose a novel learning-based framework, the neural vortex method (NVM). NVM builds a neural-network description of the Lagrangian vortex structures and their interaction dynamics to reconstruct the high-resolution Eulerian flow field in a physically-precise manner. The key components of our infrastructure consist of two networks: a vortex detection network to identify the Lagrangian vortices from a grid-based velocity field and a vortex dynamics network to learn the underlying governing interactions of these finite structures. By embedding these two networks with a vorticity-to-velocity Poisson solver and training its parameters using the fluid data obtained from grid-based numerical simulation, we can predict the accurate fluid dynamics on a precision level that was infeasible for all the previous conventional vortex methods. We demonstrate the efficacy of our method in generating highly accurate prediction results with low computational cost by predicting the evolution of the leapfrogging vortex rings system, the turbulence system, and the systems governed by Navier–Stokes (NS) equations with different external forces. We compare the prediction results made by NVM and the Lagrangian vortex method (LVM) for solving the NS equation in the periodic box and find that the relative error of the predicted velocity using NVM is more than 10 times lower than that of the LVM. Moreover, our method only requires data collected from a very short training window, more than 100 times smaller than the prediction period, which potentially facilitates data acquisition in real systems.

Quantum integrability: Lagrangian 1-form case

A new notion of integrability called the multi-dimensional consistency for the integrable systems with the Lagrangian 1-form structure is captured in the geometrical language for quantum level. A zero-curvature condition, which implies the multi-dimensional consistency, will be a key relation, e.g. Hamiltonian operators. Therefore, the existence of the zero-curvature condition directly leads to the path-independent feature of the mapping, e.g. multi-time evolution in the Schrödinger picture. Another important result is the formulation of the continuous multi-time propagator. With this new type of the propagator, a new perspective on summing all possible paths unavoidably arises as not only all possible paths in the space of dependent variables but also in the space of independent variables must be taken into account. The semi-classical approximation is applied to the multi-time propagator expressing in terms of the classical action and the fluctuation around it. Therefore, the extremum propagator, resulting in path independent feature on the space of independent variables, would guarantee the integrability of the system.

Lagrangian stretching reveals stress topology in viscoelastic flows

Viscoelastic flows are pervasive in a host of natural and industrial processes, where the emergence of nonlinear and time-dependent dynamics regulates flow resistance, energy consumption, and particulate dispersal. Polymeric stress induced by the advection and stretching of suspended polymers feeds back on the underlying fluid flow, which ultimately dictates the dynamics, instability, and transport properties of viscoelastic fluids. However, direct experimental quantification of the stress field is challenging, and a fundamental understanding of how Lagrangian flow structure regulates the distribution of polymeric stress is lacking. In this work, we show that the topology of the polymeric stress field precisely mirrors the Lagrangian stretching field, where the latter depends solely on flow kinematics. We develop a general analytical expression that directly relates the polymeric stress and stretching in weakly viscoelastic fluids for both nonlinear and unsteady flows, which is also extended to special cases characterized by strong kinematics. Furthermore, numerical simulations reveal a clear correlation between the stress and stretching field topologies for unstable viscoelastic flows across a broad range of geometries. Ultimately, our results establish a connection between the Eulerian stress field and the Lagrangian structure of viscoelastic flows. This work provides a simple framework to determine the topology of polymeric stress directly from readily measurable flow field data and lays the foundation for directly linking the polymeric stress to flow transport properties.

Unfolding a Hidden Lagrangian Structure of a Class of Evolution Equations

It is shown that a simple modification of the standard Lagrangian underlying the dynamics of Newtonian lattices enables one to infer the hidden Lagrangian structure of certain classes of first order in time evolution equations which lack the conventional Lagrangian structure. Implication to other setups is outlined and exemplified.

Gauge-Theoretic Study of Kundt Tube Experiment and Spontaneous Symmetry Transitions

In the Kundt’s experiment of acoustic resonance in closed tubes, two characteristic lengths were observed: one is the wave-length of the sound waves in resonance and the other the scale of dust striation. The latter has remained unresolved for its formation mechanism. Based on the Fluid Gauge Theory proposed recently by the author, formation mechanism of the dust striation is studied. When the sound is weak enough, the striation is unobserved. Once the wave intensity exceeds a threshold value, dust striations are formed. Formation of the dust striation is understood as a spontaneous transition of symmetry in the acoustics. According to the Theory, there is a transition of stress field within the fluid flow. Whereas the stress field is isotropic before transition, it becomes anisotropic after the transition. This is analogous to the spontaneous symmetry breaking known in the field theory. Lagrangian structures of both systems are verified to be analogous either.

On the Lagrangian structure of transport equations: Relativistic Vlasov systems

We study the Lagrangian structure of relativistic Vlasov systems, such as the relativistic Vlasov‐Poisson and the relativistic quasi‐eletrostatic limit of Vlasov‐Maxwell equations. We show that renormalized solutions of these systems are Lagrangian and that these notions of solution, in fact, coincide. As a consequence, finite‐energy solutions are shown to be transported by a global flow. Moreover, we extend the notion of generalized solution for “effective” densities, and we prove the existence of such solutions. Finally, under a higher integrability assumption of the initial condition, we show that solutions have every energy bounded, even in the gravitational case. These results extend to our setting those recently obtained for the Vlasov‐Poisson system in a series of papers by Ambrosio, Colombo, and Figalli; here, we analyze relativistic systems and also consider the contribution of the magnetic force into the evolution equation.

Sliding-Mode Control of Full-State Constraint Nonlinear Systems: A Barrier Lyapunov Function Approach

This study presents the design of a robust control based on the sliding-mode theory to solve both; the stabilization and the trajectory tracking problems of nonlinear systems subjected to a class of full-state restrictions. The selected nonlinear system satisfies a standard Lagrangian structure affected by nonparametric uncertainties. A barrier Lyapunov function is used to ensure the state constraints by designing a time-varying gain, which guarantees the fulfillment of the predefined state constraints even under external perturbations. The proposed design methodology for the barrier sliding-mode control (BSMC) ensures the convergence of the sliding surface in finite time to the origin. Consequently, the asymptotic convergence of the states to the corresponding equilibrium point is achieved. The finite-time stability of the origin in the closed-loop system with the proposed controller has been demonstrated using the second Lyapunov stability method. The suggested controller was evaluated on a two-link robotic manipulator. Then, the obtained results showed better stabilization and tracking performances (while the restrictions are satisfied) than the traditional first-order sliding-mode or linear state feedback controllers.

About the structure of the discrete and continuous Eringen’s nonlocal elastica

The Eringen’s nonlocal elastica equation does not possess a Lagrangian formulation. In this article, we find a variational integrating factor which enables us to provide a Lagrangian and Hamiltonian structure associated to this equation. Explicit expressions of the solutions in terms of elliptic integrals of the first kind are then deduced. We then derive discrete version of the Eringen’s nonlocal elastica preserving the Lagrangian and Hamiltonian structure and compare it with Challamel’s and co-worker definition of a discrete Eringen’s nonlocal elastica.

Some new aspects of fractal superconductivity

The purpose of this paper is to develop, in fractal dimension, the semi-quantitative extension of the Ginzburg-Landau theory of superconductivity. Our theoretical analysis is based on the notion of the non-standard Lagrangian approach which, based on recent studies, offer new insights in the theory of nonlinear differential equations and quantum mechanics. Besides, the fractal dimensional approach was based on the concept of the product-like fractal measure introduced by Li and Ostoja-Starzewski in order to study anisotropic media. The extended Ginzburg-Landau equation in fractal dimension was derived and several features have been revealed in particular for fractal dimension α=1/2. In particular, the theory is characterized by discrete effective mass and effective coherent length which is independent of the temperature and it don't diverge at critical temperature. This outcome is motivating since it allows manufacturing temperature-independent superconductors with a high current density without the need of refrigeration or explicit substances. The effective discrete coherent length was found to decrease with the energy levels and such a shrinking is detected in several superconductors including YBCO, Zr-doped (GdY) BCO tapes and SQUIDs. We have also studied the Abrikosov's anisotropic vortex lattice. It was found that the extended fractal theory is deformed due to the effective potential of theory which holds an inverse square part and the associated Schrödinger equation is analogous to the biconfluent Heun's differential equation. We have obtained the solution in terms of the Hermite polynomials and the energy equation was found to be altered considerably. However, it was observed that a transition between a type-II and a type-I superconductors takes place if only the non-standard parameter is ξ0≈0.99orξ0≈0.13where both numerical values result on a Ginzburg-Landau parameter κ≈1/2 although the theory is characterized by a deformed periodic lattice structure due to its anisotropic feature and non-standard Lagrangian structure. Further points are discussed and analyzed.

Numerical proof of shell model turbulence closure

The development of turbulence closure models, parametrizing the influence of small non-resolved scales on the dynamics of large resolved ones, is an outstanding theoretical challenge with vast applicative relevance. We present a closure, based on deep recurrent neural networks, that quantitatively reproduces, within statistical errors, Eulerian and Lagrangian structure functions and the intermittent statistics of the energy cascade, including those of subgrid fluxes. To achieve high-order statistical accuracy, and thus a stringent statistical test, we employ shell models of turbulence. Our results encourage the development of similar approaches for 3D Navier-Stokes turbulence.

Nonrelativistic strings and the limits of mathcal{N} = 2 dualities

We carry out the formulation of nonrelativistic (NR) F-string theory for mathcal{N} = 2 Gaiotto-Maldacena class of geometries in type IIA supergravity. Considering the NS-NS sector, we reduce the type IIA theory into the NR sigma models those are formulated over the nonrelativistic target space geometries known as the torsional String Newton-Cartan (TSNC) backgrounds. Using these TSNC data, we write down the sigma model action and explore several properties in the semiclassical limit. In particular, we carry out the canonical analysis of the T-duality transformation rules in TSNC background. Following the standard canonical prescription of obtaining T-duality transformations, we map these TSNC sigma models to relativistic sigma models propagating over general relativity backgrounds. Finally, we conclude with a detailed discussion on the effects of adding flavour branes and thereby taking the corresponding TSNC limit in the bulk description of mathcal{N} = 2 quivers. We compute various field theory observables in the nonrelativistic limit and outline the structure of the NR Lagrangian following the light cone reduction of mathcal{N} = 2 SYM theory.

Transitional flow structures in heated hypersonic boundary layers

Transition in a Mach 6 flared cone boundary layer over a heated wall has been investigated in the Mach 6 wind tunnel at Peking University using visualization, focused laser differential interferometry, infrared imaging, particle image velocimetry (PIV), and direct numerical simulation (DNS). The model's wall-temperature ratio η=Tw/T0 (where Tw and T0 are, respectively, the wall temperature and oncoming stream stagnation temperature) can be controlled to vary from 0.66 to 1.77. An ultrafast illumination image system has been used for Rayleigh-scattering visualization and PIV to experimentally capture the dynamics of the transition. Lagrangian flow structures are revealed by both the DNS results and the time-resolved PIV data. The effect of wall temperature on the transition is investigated, and it is found that increasing η initially delays but then promotes the transition to turbulence, with the reversal point being near η≈1. The turbulence onset mechanism over the heated wall for η=1.50, where first-mode-induced oblique breakdown dominates, is then investigated, and it is shown that lifting-up three-dimensional (3D) waves appear around the critical layer owing to the nonlinear development of the oblique first mode. Consequently, a downward sweep motion occurs to compensate for the lifting low-speed fluid, with the formation of a warped wave front. High-shear layers are created around the 3D Lagrangian waves and strengthened to cause the formation of a Λ-vortex. In general, this lifting-up 3D wavepacket has been confirmed to play a determining role in hypersonic turbulence production over a heated wall, which is similar to the findings in incompressible boundary layers.

Moduli of flat connections on smooth varieties

We study the moduli functor of flat bundles on smooth, possibly non-proper, algebraic variety $X$ (over a field of characteristic zero). For this we introduce the notion of \emph{formal boundary} of $X$, denoted by $\partial X$, which is a formal analogue of the boundary at infinity of the Betti topological space associated to $X$. We explain how to construct two derived moduli functors $Vect^{\nabla}(X)$ and $Vect^{\nabla}(\partial X)$, of flat bundles on $X$ and on $\partial X$, as well as a restriction map $R : Vect^{\nabla}(X) \rightarrow Vect^\nabla(\partial X)$ from the former to the later. This work contains two main results. First we prove that the morphism R comes equipped with a canonical shifted Lagrangian structure in the sense of [PTVV]. This first result can be understood as the de Rham analogue of the existence of Poisson structures on moduli of local systems previously studied by the authors. As a second statement, we prove that the geometric fibers of $R$ are representable by quasi-algebraic spaces, a slight weakening of the notion of algebraic spaces.

Construction of the differential equations system of the program motion in Lagrangian variables in the presence of random perturbations

The classification of inverse problems of dynamics in the class of ordinary differential equations is given in the Galiullin’s monograph. The problem studied in this paper belongs to the main inverse problem of dynamics, but already in the class of second-order stochastic differential equations of the Ito type. Stochastic equations of the Lagrangian structure are constructed according to the given properties of motion under the assumption that the random perturbing forces belong to the class of processes with independent increments. The problem is solved as follows: First, a second-order Ito differential equation is constructed so that the properties of motion are the integral manifold of the constructed stochastic equation. At this stage, the quasi-inversion method, Erugin’s method and Ito’s rule of stochastic differentiation of a complex function are used. Then, by applying the constructed Ito equation, an equivalent stochastic equation of the Lagrangian structure is constructed. The necessary and sufficient conditions for the solvability of the problem of constructing the stochastic equation of the Lagrangian structure are illustrated by the example of the problem of constructing the Lagrange function from a motion property of an artificial Earth satellite under the action of gravitational forces and aerodynamic forces.

From underactuation to quasi‐full actuation: Aiming at a unifying control framework for articulated soft robots

AbstractWe establish a structure preserving state and input transformation that allows a class of underactuated Euler Lagrange systems to be treated as “quasi‐fully” actuated. In this equivalent quasi‐fully actuated form, the system is characterized by the same Lagrangian structure as the original one. This facilitates the design of control approaches that take into account the underlying physics of the system and that shape the system dynamics to a minimum extent. Due to smoothness constraints on the new input vector that acts directly on the noncollocated coordinates, we coin the term quasi‐fully actuated. The class of Euler–Lagrange systems we consider is the class of articulated soft robots with nonlinear spring characteristics that are modeled with a block diagonal inertia matrix. We illustrate how the quasi‐fully actuated form enables the direct transfer of control concepts that have been derived for fully actuated manipulators. We adopt the popular energy‐shaping and two passivity‐based concepts. The exemplary adoptions of the PD+ and Slotine and Li controllers allow us to solve the task‐space tracking problem for highly elastic joint robots with nonlinear spring characteristics. These control schemes allow compliant behavior of the robot's TCP to be specified with respect to a reference trajectory. A key aspect of the presented framework is that it enables the adoption of rigid joint controllers as well as concepts underlying the original stability analysis. We believe that our framework presents an important step toward unifying the control design for rigid and articulated soft robots.

Conservative, density-based smoothed particle hydrodynamics with improved partition of the unity and better estimation of gradients

Context. The accurate evaluation of gradients is a cornerstone of the smoothed particle hydrodynamics (SPH) technique. Using an integral approach to estimating gradients has been proven to substantially enhance its accuracy, retaining the Lagrangian structure of SPH equations and remaining fully conservative. However, in practice, it is difficult to ensure that the Lagrangian formulation is entirely consistent with regard to the exact partition of the unity. Aims. In this paper, we focus our study on the connection between the choice of the volume elements (VEs) in the SPH summations as well as the accuracy in the gradient estimation within the integral approach scheme (ISPH). We propose a new variant of VEs to improve the partition of the unity that is fully compatible with the Lagrangian formulation of SPH, including grad-h corrections. Aims. Using analytic considerations, simple static toy models in 1D, and a set of full 3D test cases, we show that any improvement in the partition of the unity also leads to a better calculation of gradients when the integral approach is used jointly. Additionally, we propose an easy-to-implement modification of the ISPH scheme, which makes it more flexible and better suited to handling sharp density contrasts. Methods. The ISPH code that is built with the proposed scheme has been validated with a good number of standard tests, some of them involving contact discontinuities. The performance of the code was shown to be excellent in all of these tests, consistently demonstrating that an improvement in the partition of the unity is not detrimental to the optimal conservation of energy, momentum, and entropy that is typical of Lagrangian schemes. Results. We successfully built a new ISPH scheme on a Lagrangian basis, which is fully conservative, and compatible with self-consistent grad-h terms and an improved partition of the unity. The ensuing code is able to successfully cope with the tensile instability and has been validated with a number of hydrodynamic tests with good results.

Lagrangian nonlocal nonlinear Schrödinger equations

In 2013 Ablowitz and Musslimani (AM) obtained a nonlocal generalization of the NLS equation which is integrable, but does not have a standard Lagrangian structure. The present paper shows that there exist two new nonlocal NLS equations, similar to the AM model, which do possess Lagrangian structures. We show that these two models (LN1 and LN2) possess solitary wave solutions which remain trapped in the neighborhood of the origin (x=0), and solitary waves which are able to escape from the origin. These two types of solutions are obtained by direct numerical solutions, and also by a variational method. In the case of LN2 model, the variational approach explains the existence of these two types of solutions. Collisions of breathers which obey the equation LN2 are numerically studied, and the results show that these breathers are robust solutions.