A system of coordinated autonomous robots for Lagrangian studies of microbes in the oceanic deep chlorophyll maximum

The deep chlorophyll maximum (DCM) layer is an ecologically important feature of the open ocean. The DCM cannot be observed using aerial or satellite remote sensing; thus, in situ observations are essential. Further, understanding the responses of microbes to the environmental processes driving their metabolism and interactions requires observing in a reference frame that moves with a plankton population drifting in ocean currents, i.e., Lagrangian. Here, we report the development and application of a system of coordinated robots for studying planktonic biological communities drifting within the ocean. The presented Lagrangian system uses three coordinated autonomous robotic platforms. The focal platform consists of an autonomous underwater vehicle (AUV) fitted with a robotic water sampler. This platform localizes and drifts within a DCM community, periodically acquiring samples while continuously monitoring the local environment. The second platform is an AUV equipped with environmental sensing and acoustic tracking capabilities. This platform characterizes environmental conditions by tracking the focal platform and vertically profiling in its vicinity. The third platform is an autonomous surface vehicle equipped with satellite communications and subsea acoustic tracking capabilities. While also acoustically tracking the focal platform, this vehicle serves as a communication relay that connects the subsea robot to human operators, thereby providing situational awareness and enabling intervention if needed. Deployed in the North Pacific Ocean within the core of a cyclonic eddy, this coordinated system autonomously captured fundamental characteristics of the in situ DCM microbial community in a manner not possible previously.

The Noether–Bessel-Hagen symmetry approach for dynamical systems

The Noether–Bessel-Hagen theorem can be considered a natural extension of Noether Theorem to search for symmetries. Here, we develop the approach for dynamical systems introducing the basic foundations of the method. Specifically, we establish the Noether–Bessel-Hagen analysis of mechanical systems where external forces are present. In the second part of the paper, the approach is adopted to select symmetries for a given systems. In particular, we focus on the case of harmonic oscillator as a testbed for the theory, and on a cosmological system derived from scalar–tensor gravity with unknown scalar-field potential [Formula: see text]. We show that the shape of potential is selected by the presence of symmetries. The approach results particularly useful as soon as the Lagrangian of a given system is not immediately identifiable or it is not a Lagrangian system.

Bipartite Consensus Control for a Swarm of Robots

This paper studies the bipartite consensus problem of a swarm of robots whose dynamics are formulated by Lagrangian equations. Two distributed bipartite consensus control protocols are proposed for a swarm of robots without a leader or with a virtual leader. For the nonleader case, the networked Lagrangian system can reach static bipartite consensus under the control protocol developed, and the final convergent states can be explicitly determined by the specific structure of the Laplacian matrix associated with the cooperative–competitive network topology. For the virtual leader case, all the followers can track the leader's state in a bipartite formation to realize bipartite tracking consensus. Finally, the simulation results are given to verify the theoretical results.

On New Analytic Free Vibration Solutions of Doubly Curved Shallow Shells by the Symplectic Superposition Method Within the Hamiltonian-System Framework

This study presents a first attempt to explore new analytic free vibration solutions of doubly curved shallow shells by the symplectic superposition method, with focus on non-Lévy-type shells that are hard to tackle by classical analytic methods due to the intractable boundary-value problems of high-order partial differential equations. Compared with the conventional Lagrangian-system-based expression to be solved in the Euclidean space, the present description of the problems is within the Hamiltonian system, with the solution procedure implemented in the symplectic space, incorporating formulation of a symplectic eigenvalue problem and symplectic eigen expansion. Specifically, an original problem is first converted into two subproblems, which are solved by the above strategy to yield the symplectic solutions. The analytic frequency and mode shape solutions are then obtained by the requirement of the equivalence between the original problem and the superposition of subproblems. Comprehensive results for representative non-Lévy-type shells are tabulated or plotted, all of which are well validated by satisfactory agreement with the numerical finite element method. Due to the strictness of mathematical derivation and accuracy of solution, the developed method provides a solid approach for seeking more analytic solutions.

On Periodic Solutions to Constrained Lagrangian System

A Lagrangian system is considered. The configuration space is a non-compact manifold that depends on time. A set of periodic solutions has been found.

Minimal Boundaries in Tonelli Lagrangian Systems

We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, that is, a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $\textrm{T} M$, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base $M$. Finally, we prove the existence of action minimizing simple periodic orbits on energies just above the Mañé critical value of the universal abelian cover. This provides in particular a class of nonreversible Finsler metrics on the two-sphere possessing infinitely many closed geodesics.