We present a smoothed particle hydrodynamic model for incompressible fluids. As opposed to solving a pressure Poisson equation in order to get a divergence-free velocity field, here incompressibility is achieved by requiring as a kinematic constraint that the volume of the fluid particles is constant. We use Lagrangian multipliers to enforce this restriction. These Lagrange multipliers play the...

The paper studies the “Lagrangian temperature” defined through the entropy maximization in the canonical ensemble, which is the negative inverse Lagrangian multiplier corresponding to the constraint of internal energy. The Lagrangian temperature is derived for systems out of thermal equilibrium described by kappa distributions such as space plasmas. The physical meaning of temperature is manife...

known as the inverse problem of the calculus of variations for the system (2.1). For historical reasons, we also refer to this problem as the Sonin–Douglas problem; Eq. (2.2) are called theSonin–Douglas equations. Having inmind the correspondence with classicalmechanics,we sometimes call the system F = (F1, . . . , Fm) the force. The functions g jk are called variational multipliers. If the fun...

Gauge fields are special in the sense that they are invariant under gauge transformations and " ipso facto " they lead to problems when we try quantizing them straightforwardly. To circumvent this problem we need to specify a gauge condition to fix the gauge so that the fields that are connected by gauge invariance are not overcounted in the process of quantization. The usual way we do this in ...

This paper presents a formation modelling scheme based on a set of inter-body constraint functions and Lagrangian multipliers. Formation control for a fleet of marine craft is achieved by stabilizing the auxiliary constraints such that the desired formation configuration appears. In the proposed framework we develop robust control laws for marine surface vessels to counteract unknown, slowly va...

This paper presents a transmission-constrained unit commitment method using a Lagrangian relaxation approach. Based on a DC power flow model, the transmission constraints are formulated as linear constraints. The transmission constraints, as well as the demand and spinning reserve constraints, are relaxed by attaching Lagrange multipliers. A three-phase algorithmic scheme is devised including d...

Recent applications of Statistics often leads one to encounter testing problems where the original hypothesis of interest comprises of the union of several sub-hypothesis. In the framework of such Intersection-Union testing of hypothesis, in contrast to the usual Union-Intersection framework, a subhypothesis therein may specify a parameter or a function of some of the parameters of the underlyi...

This paper presents a dynamic model of a planar parallel manipulator including structural flexibility of several linkages. The equations of motion are formulated using the Lagrangian equations of the first type and Lagrangian multipliers are introduced to represent the geometry of multiple closed loop chains. Then, an active damping approach using a PZT actuator is described to attenuate struct...

Safe reinforcement learning considers practical scenarios that maximize the return while satisfying safety constraints. Current algorithms, which suffer from training oscillations or approximation errors, still struggle to update policy efficiently with precise constraint satisfaction. In this article, we propose Augmented Proximal Policy Optimization (APPO), augments Lagrangian function of pri...

An extension of the Legendre transform to non-convex functions with vanishing Hessian as a mix of envelope and general solutions of the Clairaut equation is proposed. Applying this to constraint systems, the procedure of finding a Hamiltonian for a singular Lagrangian is just that of solving a corresponding Clairaut equation with a subsequent application of the proposed Legendre-Clairaut transf...