Lagrangian Structures Research Articles

Regular Lagrangian flow for wavelike vector fields and the Vlasov-Maxwell system

In this paper, we study the Lagrangian structure of Vlasov-Maxwell system, that is, by using a suitable notion of flow, we prove that if the densities ρ,j are integrable in spacetime, and the charge acceleration ∂tj and ∂ttj (or ∇∂tj) are integrable functions in spacetime, then renormalized and distributional solutions of the system are the transport of the initial condition by its flow. We study more general vector fields, with wavelike structure in the sense that it has finite speed of propagation, generalizing the vector fields studied in [6]. The result is a extension of those obtained by Ambrosio, Colombo, and Figalli [2] for the Vlasov-Poisson system, and by the author and Marcon [5] for relativistic Vlasov-systems with quasistatic approximations of Maxwell's equations.

Levinson–Smith Dissipative Equations and Geometry of GENERIC Formalism and Contact Hamiltonian Mechanics

We apply Jacobi’s Last Multiplier theory to construct the non-standard Lagrangian and Hamiltonian structures for the Levinson–Smith equations satisfying the Chiellini integrability condition. Then after a brief exposition of the contact geometry, we explore its connection with the non-standard Hamiltonian structures. We present the formulation of the Levinson–Smith equation in terms of General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) method and also study the gradient-type flow. We give a geometric formulation of GENERIC and apply this to general Levinson–Smith equations.

Lagrangian multiform structure of discrete and semi-discrete KP systems

A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semi-discrete variants including several differential-difference equations asssociated with, and compatible with, the partial difference equation. To this end, an overview is given of the various (discrete and semi-discrete) variants of the KP system, and their associated Lax representations, including a novel `generating PDE' for the KP hierarchy. The exterior derivative of the Lagrangian 3-form for the lattice potential KP equation is shown to exhibit a double-zero structure, which implies the corresponding generalised Euler-Lagrange equations. Alongside the 3-form structures, we develop a variational formulation of the corresponding Lax systems via the square eigenfunction representation arising from the relevant direct linearization scheme.

Finite element methods for multicomponent convection-diffusion

Abstract We develop finite element methods for coupling the steady-state Onsager–Stefan–Maxwell (OSM) equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization and extensibility to the transient, anisothermal and nonideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the definition of mass-average velocity in the OSM equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting structure mandated by linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to nonideal fluids with a simulation of the microfluidic mixing of hydrocarbons.

Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale dynamical systems

This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Lagrangian systems, which includes nonlinear wave equations. Existing intrusive projection-based model reduction approaches construct structure-preserving Lagrangian ROMs by projecting the Euler–Lagrange equations of the full-order model (FOM) onto a linear subspace. This Galerkin projection step requires complete knowledge about the Lagrangian operators in the FOM and full access to manipulate the computer code. In contrast, the proposed Lagrangian operator inference approach embeds the mechanics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying Lagrangian structure. The proposed approach exploits knowledge of the governing equations (but not their discretization) to define the form and parametrization of a Lagrangian ROM which can then be learned from projected snapshot data. The method does not require access to FOM operators or computer code. The numerical results demonstrate Lagrangian operator inference on an Euler–Bernoulli beam model, the sine-Gordon (nonlinear) wave equation, and a large-scale discretization of a soft robot fishtail with 779,232 degrees of freedom. The learned Lagrangian ROMs generalize well, as they can accurately predict the physical solutions both far outside the training time interval, as well as for unseen initial conditions.

Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems

Complex mechanical systems often exhibit strongly nonlinear behavior due to the presence of nonlinearities in the energy dissipation mechanisms, material constitutive relationships, or geometric/connectivity mechanics. Numerical modeling of these systems leads to nonlinear full-order models that possess an underlying Lagrangian structure. This work proposes a Lagrangian operator inference method enhanced with structure-preserving machine learning to learn nonlinear reduced-order models (ROMs) of nonlinear mechanical systems. This two-step approach first learns the best-fit linear Lagrangian ROM via Lagrangian operator inference and then presents a structure-preserving machine learning method to learn nonlinearities in the reduced space. The proposed approach can learn a structure-preserving nonlinear ROM purely from data, unlike the existing operator inference approaches that require knowledge about the mathematical form of nonlinear terms. From a machine learning perspective, it accelerates the training of the structure-preserving neural network by providing an informed prior (i.e., the linear Lagrangian ROM structure), and it reduces the computational cost of the network training by operating on the reduced space. The method is first demonstrated on two simulated examples: a conservative nonlinear rod model and a two-dimensional nonlinear membrane with nonlinear internal damping. Finally, the method is demonstrated on an experimental dataset consisting of digital image correlation measurements taken from a lap-joint beam structure from which a predictive model is learned that captures amplitude-dependent frequency and damping characteristics accurately. The numerical results demonstrate that the proposed approach yields generalizable nonlinear ROMs that exhibit bounded energy error, capture the nonlinear characteristics reliably, and provide accurate long-time predictions outside the training data regime.

On the Lagrangian multiform structure of the extended lattice Boussinesq system

The lattice Boussinesq (lBSQ) equation is a member of the lattice Gel'fand-Dikii (lGD) hierarchy, introduced in \cite{NijPapCapQui1992}, which is an infinite family of integrable systems of partial difference equations labelled by an integer $N$, where $N=2$ represents the lattice Korteweg-de Vries (KdV) system, and $N=3$ the Boussinesq system. In \cite{Hiet2011} it was shown that, written as three-component system, the lBSQ system allows for extra parameters which essentially amounts to building the lattice KdV inside the lBSQ. In this paper we show that, on the level of the Lagrangian structure, this boils down to a linear combination of Lagrangians from the members of the lGD hierarchy as was established in \cite{LobbNijGD2010}. The corresponding Lagrangian multiform structure is shown to exhibit a `double zero' structure.

Manifestly covariant polynomial M5-brane lagrangians

We present polynomial and manifestly covariant M5-brane Lagrangians along with their analyses involving their dynamics, gauge symmetries and their nonlinear self-duality condition. Such Lagrangians can be particularly useful for developments that are otherwise hindered by a non-polynomial structure and singularity of the Lagrangian such as its quantisation. Although on integrating out some of the auxiliary fields these polynomial Lagrangians reduce to the M5-brane Lagrangian given by the Pasti-Sorokin-Tonin (PST) formalism, in the analysis of the polynomial Lagrangians the only remnant of the non-polynomial structure of the PST type Lagrangian appears in the gauge transformation corresponding to an infinitesimal shift of a Stückelberg field. This transformation does not affect the dynamics or the on-shell self-duality condition of the polynomial M5-brane Lagrangians.

Variational and Dirac structures for interconnected distributed-discrete systems

We present how the interconnection of distributed and discrete systems can be realized within the setting of variational principles and Dirac structures on the Lagrangian side. For the distributed system we focus on the case when the configuration space is an infinite-dimensional vector space and the system is subject to boundary energy flow. A key property of our approach is that it systematically extends the canonical variational and symplectic structures of finite and infinite-dimensional Lagrangian and Hamiltonian mechanics.

Construction of a functional by a given second-order Ito stochastic equation

Abstract In this article, we consider the problem of extending Hamilton’s principle to the class of natural mechanical systems with random perturbing forces of white noise type. By the method of moment functions, we construct the functionals taking a stationary value on the solutions of a given stochastic equation of Lagrangian structure.

Canonical curves and Kropina metrics in Lagrangian contact geometry

We present a Fefferman-type construction from Lagrangian contact to split-signature conformal structures and examine several related topics. In particular, we describe the canonical curves and their correspondence. We show that chains and null-chains of an integrable Lagrangian contact structure are the projections of null-geodesics of the Fefferman space. Employing the Fermat principle, we realize chains as geodesics of Kropina (pseudo-Finsler) metrics. Using recent rigidity results, we show that ‘sufficiently many’ chains determine the Lagrangian contact structure. Separately, we comment on Lagrangian contact structures induced by projective structures and the special case of dimension three.

Spatiotemporal evolution of particle puffs in transitional channel flow

We study the shape evolution of puffs composed of tracers advected in a transitional channel flow. We perform a direct numerical simulation of a spatially evolving channel flow, where the inflow condition is given by a solution of the Orr–Sommerfeld equation and the flow evolves through all stages of transition up to fully developed turbulence. In this setting, we release spherical puffs of particles and track their evolution using measures derived from their approximation as ellipsoids. By varying the initial position of puffs, we characterize the spatial non-homogeneity of the flow, with respect to both the distance from the wall and, most importantly, the streamwise coordinate along which the flow evolves. Furthermore, we assess the influence of scale-dependent phenomena on puff shapes by varying the initial size of the clouds of particles. The present approach explores the interaction between flow features and advected Lagrangian structures. Additionally, it reveals the interplay between flow scales and how their balance changes during transition, where the intermittency causes large puffs to be much less elongated than smaller puffs independently of the distance from the walls.

Saturation and Multifractality of Lagrangian and Eulerian Scaling Exponents in Three-Dimensional Turbulence.

Inertial-range scaling exponents for both Lagrangian and Eulerian structure functions are obtained from direct numerical simulations of isotropic turbulence in triply periodic domains at Taylor-scale Reynolds number up to 1300. We reaffirm that transverse Eulerian scaling exponents saturate at ≈2.1 for moment orders p≥10, significantly differing from the longitudinal exponents (which are predicted to saturate at ≈7.3 for p≥30 from a recent theory). The Lagrangian scaling exponents likewise saturate at ≈2 for p≥8. The saturation of Lagrangian exponents and transverse Eulerian exponents is related by the same multifractal spectrum by utilizing the well-known frozen hypothesis to relate spatial and temporal scales. Furthermore, this spectrum is different from the known spectra for Eulerian longitudinal exponents, suggesting that Lagrangian intermittency is characterized solely by transverse Eulerian intermittency. We discuss possible implications of this outlook when extending multifractal predictions to the dissipation range, especially for Lagrangian acceleration.

Adaptive controller based on barrier Lyapunov function for a composite Cartesian-delta robotic device for precise time-varying position tracking

This study presents the design of an adaptive event-driven controller for solving the trajectory tracking problem of a composite robotic device made up of a three-dimensional Cartesian and a parallel Delta robot. The proposed composite device has a mathematical model satisfying a standard Lagrangian structure affected by modeling uncertainties and external perturbations. The adaptive gain of the controller is considered to enforce the convergence of the tracking error while the state bounds are satisfied. The barrier Lyapunov function addresses the preconceived state constraints for both robotic devices by designing a time-varying gain that guarantees the ultimate boundedness of the tracking error under the effect of external perturbations. The event-driven approach considers that the Cartesian robot is moving into a predefined invariant zone near to the origin. In contrast, the delta robot can complete the tracking problem once the end-effector is inside the given zone. The suggested controller was evaluated using a virtual representation of the composite robotic device showing better tracking performance (while the restrictions are satisfied) than the performances obtained with the traditional linear state feedback controllers. Analyzing the mean square error and its integral led to confirming the benefits of using the adaptive barrier control.

Lagrangian large eddy simulations via physics-informed machine learning

High-Reynolds number homogeneous isotropic turbulence (HIT) is fully described within the Navier-Stokes (NS) equations, which are notoriously difficult to solve numerically. Engineers, interested primarily in describing turbulence at a reduced range of resolved scales, have designed heuristics, known as large eddy simulation (LES). LES is described in terms of the temporally evolving Eulerian velocity field defined over a spatial grid with the mean-spacing correspondent to the resolved scale. This classic Eulerian LES depends on assumptions about effects of subgrid scales on the resolved scales. Here, we take an alternative approach and design LES heuristics stated in terms of Lagrangian particles moving with the flow. Our Lagrangian LES, thus L-LES, is described by equations generalizing the weakly compressible smoothed particle hydrodynamics formulation with extended parametric and functional freedom, which is then resolved via Machine Learning training on Lagrangian data from direct numerical simulations of the NS equations. The L-LES model includes physics-informed parameterization and functional form, by combining physics-based parameters and physics-inspired Neural Networks to describe the evolution of turbulence within the resolved range of scales. The subgrid-scale contributions are modeled separately with physical constraints to account for the effects from unresolved scales. We build the resulting model under the differentiable programming framework to facilitate efficient training. We experiment with loss functions of different types, including physics-informed ones accounting for statistics of Lagrangian particles. We show that our L-LES model is capable of reproducing Eulerian and unique Lagrangian turbulence structures and statistics over a range of turbulent Mach numbers.

The wavedrifter: a low-cost IMU-based Lagrangian drifter to observe steepening and overturning of surface gravity waves and the transition to turbulence

ABSTRACT Waves and wave breaking are important to many deep and shallow water processes. We describe the wavedrifter, an in situ water-following inertial measurement unit (IMU)-based drifter that measures wave steepening and overturning kinematics, and the subsequent transition to turbulence. The wavedrifter has 5 cm diameter, 77 g mass, and 0.84 saltwater specific gravity. GPS provides time synchronization. MATLAB’s Attitude, Heading, Reference System (AHRS) library provides wavedrifter orientation. Laboratory experiments quantify the wavedrifter vertical oscillation mode, water following properties, and ability to reproduce wave spectra for small, f = 1.5 Hz waves. The wavedrifter observed wave overturning and the transition to turbulence at the Surf Ranch wave basin. Synchronized video observations provide context. The upper-back of the overturn had large ( ≈ 4 g ) accelerations and 14 g acceleration magnitude occurs with the impact of the overturning jet. Trajectories reveal the Lagrangian structure of the overturn and subsurface vortex. Although it has limitations, the wavedrifter is a powerful in situ tool to probe wave processes.

The Lagrangian structure, the Euler equation, and second Newton’s law of ultrafast nonlinear optics

With a notable exception of space–time variable swap, pulse evolution equations of ultrafast optics are mathematically similar to the equations of quantum mechanics and beam dynamics. The Lagrangian structure of these equations is, however, much less intuitive. Here, we show that such a structure can be identified via a path-integral analysis of the pulse evolution equation, offering useful insights into the Lagrangian underpinning of ultrafast nonlinear optics. The Euler–Lagrange equation applied to the optical analog of the Lagrange function leads to a second-order differential equation that parallels in its structure second Newton’s law. Stationary-phase space–time paths found by solving such an equation reveal the hidden inner workings behind a vast class of field-waveform transformations in ultrafast optics, including the buildup of nonlinear phase, modulation instabilities, and soliton breather dynamics.

On the stability of Yang-Mills vacuum

The (anti)self-dual covariantly constant vacuum fields have positive/stable and infinitely many zero modes, so called Leutwyller chromons. A regularisation method is suggested that allows to sum the contribution of the zero modes and to obtain the effective Lagrangian that has contribution of all positive/stable modes and zero modes. In the second approach we integrated over the collective coordinates of the nonlinearly interacting zero modes and obtained the same result. The last method is a generalisation of the t'Hooft integration over the instanton zero mode collective coordinates. It is important that both methods lead to the same result indicating the robustness of the logarithmic structure of the effective Lagrangian.

Quantitative analysis of enhanced mixing and combustion by lobed mixer in a ramjet engine: Study using hyperbolic Lagrangian coherent structures

Mixing and combustion are the core processes in the operation of the ramjet engine, and the contact line length (2D) or contact area (3D) is an important bridge to analyze the link between mixing and combustion. In this paper, a new formula for calculating the contact area of two components in an arbitrary complex region is proposed, which can analyze the dynamic mixing process of the two components. It can be used to evaluate the mixing efficiency in the dynamic mixing process, the larger the contact area, the higher the mixing efficiency. This paper establishes a new method to evaluate the mixing in ramjet engines based on the Lagrangian coherence structures, and uses this method to study the feasibility of the application of the lobed mixer in ramjet engines. The study shows that the vortex induced by the lobed mixer can significantly increase the contact area between gas and air in the combustion chamber, effectively enhancing the mixing and combustion of gas and air therein. The average mixing efficiency during the operation increased by 11.046% and the combustion efficiency increased by 8.318%.

Galilei-invariant and energy-preserving extensions of Benjamin–Bona–Mahony-type equations

The classical Benjamin–Bona–Mahony equation (BBM equation) models unidirectional propagation of long gravity surface waves of small amplitude. Unlike many other water wave models, it lacks the Galilean invariance, which is an essential property of physical systems. It is shown that by an addition of a higher asymptotic order nonlinear term, this deficiency can be corrected, giving rise to a new Galilei invariant Benjamin–Bona–Mahony equation (iBBM equation). Moreover, further additional higher-order terms can be chosen in a way that the augmented model preserves the energy conservation property along with Hamiltonian and Lagrangian structures. The resulting equation is referred to as energy-preserving Benjamin–Bona–Mahony equation (eBBM).It is shown that both the classical BBM equation and the energy-preserving eBBM equations belong to a one-parameter (α) family that shares essentially the same local and nonlocal symmetries, conservation laws, Hamiltonian, and Lagrangian structures, with the BBM and eBBM equations corresponding to parameter values α=0 and α=1, respectively. Symmetry and conservation law classifications reveal a special case α=1/3, which is shown to correspond to a rescaled version of the celebrated integrable Camassa–Holm (CH) equation. Local symmetries and conservation laws are computed, and numerical solution behaviour is compared for the three BBM-type modes and the CH-equivalent eBBM1/3 model.