Elastica as a dynamical system

Bates, Larry; Chhabra, Robin; Śniatycki, Jędrzej

doi:10.1016/j.geomphys.2016.09.001

Cited by 2 publications

(5 citation statements)

References 13 publications

(19 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 5. It is clear from the above proof that Theorem 4 is a generalization of known results, e.g., see Bates et al [23], in particular, Theorem A.31 and Remark A.32. Note that equations (45) and ( 46) have the same left-hand side (in the coordinates defined by equation ( 39)) as that of Bates et al (Theorem A.31 and Remark A.32), but the right-hand side is different.…”

Section: T -Lagrangian: Second-order Lagrangianssupporting

confidence: 53%

“…Using equation (39) (for n = 2) and the above remark, the Lagrangian of this functional (integrand of equation ( 7)) can be written as L e = (η/r) 2 r, which agrees with form suggested by the representation Theorem 4. Note that equation ( 7) is parametrization invariant; an observation has also been made recently in [23]. According to Corollary 1, such functionals are T -Lagrangians.…”

Section: T -Lagrangian: Second-order Lagrangiansmentioning

confidence: 63%

“…As a counterpart of above statement for a second-order Lagrangian [23] (e.g., see Theorem A.31 and Remark A.32), we mention Theorem 2.…”

Section: Revisiting Parametrization Invariancementioning

confidence: 91%

“…Altogether these weakened notions of invariance are anticipated to involve a significant role played by null Lagrangians. Regarding the ways of this paper, the simplicity of presented one-dimensional setting also allows a possible analysis on lines of Bates et al [23] that exploits jet bundle formalism; it is expected that the same formalism becomes handy and has strong potential in higher dimensions where our approach becomes tedious.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Generalizing parametrization invariance in the calculus of variations

Dharmavaram

Sharma

2023

Mathematics and Mechanics of Solids

View full text Add to dashboard Cite

We revisit the notion of parametrization invariance while introducing certain weakened notions of invariance in the calculus of variations. In this work, we employ a straightforward approach in the classical setting and mostly restrict attention to functionals on one-dimensional domains. We establish a connection between parametrization invariant functionals and functionals embodying a weaker notion of invariance of their Lagrangian; we term this notion as [Formula: see text]- Lagrangian analogous to the well-known idea of null Lagrangian. However, the Euler–Lagrange operator of a [Formula: see text]-Lagrangian vanishes only along the tangential direction in the configuration space. On one-dimensional domain and for first- and second-order theories, we show that functional described by such a Lagrangian is necessarily a parametrization invariant functional modulo null Lagrangian. Keeping the motivation for partial differential equations, we also introduce and explore the notion of [Formula: see text]- Lagrangian, with an invariance complementary to the case of [Formula: see text]- Lagrangian, whose Euler–Lagrange operator vanishes along normal directions. We find that in a one-dimensional setting, every [Formula: see text]-Lagrangian is simply a null Lagrangian.

show abstract

Section: T -Lagrangian: Second-order Lagrangianssupporting

confidence: 53%

Section: T -Lagrangian: Second-order Lagrangiansmentioning

confidence: 63%

“…As a counterpart of above statement for a second-order Lagrangian [23] (e.g., see Theorem A.31 and Remark A.32), we mention Theorem 2.…”

Section: Revisiting Parametrization Invariancementioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalizing parametrization invariance in the calculus of variations

Dharmavaram

Sharma

2023

Mathematics and Mechanics of Solids

View full text Add to dashboard Cite

show abstract

“…Continuum Mechanics have also been used with beam and rod theory to formulate PDEs that fully capture the infinite dimensional deformation of flexible bodies [63][64][65]. 1D models of have been shown sufficient for modelling dynamic response, and have even been derived on Lie groups [66]. The earliest attempts introduced Euler-Bernoulli beam theory to capture the bending of rigid bodies and then Timoshenko models to also capture shear effects but they are both limited to small deformations [46].…”

Section: Continuum Modelsmentioning

confidence: 99%

Geometric Recursive Dynamic Modelling and Simulation of Soft Robotic Systems

Samei

View full text Add to dashboard Cite

This thesis studies the dynamics of soft robotic systems modelled as rigid-flexible multi-body systems. A numerical framework is developed to capture finite deformations including extension, shear, bending and torsion of a dynamic 1-Dimensional (1D) flexible body on the Special Euclidean group SE(3) based on Cosserat rod theory. The governing parameterization-free equations are expressed as a set of Partial Differential Equations (PDEs) on the SE(3). An implicit differentiation method semidiscretizes the time derivatives on the Lie algebra of SE(3) to convert the PDEs to a set of Ordinary Differential Equations (ODEs) in the arc-length of the continuum. The novelty of this work is the implementation of a finite difference solution on SE(3) along with a higher-order Runge-Kutta-Munthe-Kaas geometric integrator employed to spatially propagate the configuration of the rod. The resulting solution is a dynamic model of the flexible body that is parameterization-free, computationally inexpensive and can be used in real-time applications.A recursive parameterization-free formulation for the forward and inverse dynamics of multi-body systems is expressed on the SE(3), using the previously developed model for flexible bodies. The system is composed of bodies serially connected with single degree of freedom actuated joints from a fixed base. The Newton-Euler equation of motion for a rigid body and a set of PDEs for a dynamic Cosserat rod are coupled to recursively formulate the dynamics of multi-body systems. The joint kinematics is captured through the exponential map of the SE(3). The inverse dynamics algorithm recursively determines the system response and the joint torques needed to follow a given joint space trajectory, while the forward dynamics algorithm determines the system's motion given joint torques. A shooting-method-based Boundary Condition (BC) solver is developed to solve for the BCs in the set of PDEs and implement a finite difference solution. This study can be implemented to develop joint-space computed-torque control strategies.Recent geometric methods for rigid-flexible multi-body dynamics often use shape This thesis is submitted under only my name but the hardest part of this endeavor, keeping me motivated, was done by everyone but me. Any progress that I have made, and more that I will, is from the guidance of my professors with a special gratitude for Prof. Chhabra, Benavides, Gerrick and Hill. As with solving any problem, running into challenges was inevitable; and in such times working on this thesis, I have relied on my friends Dana, Paige, Wissam, Marana, Harry and Zach to remind me what it means to love learning. More often than not, I have sustained an unhealthy fixation on my studies. So I am grateful to my colleagues Reza, Vaughn, Aman, Ini, Kamala and Sara for taking better care of me than I did of myself and my homies Elias, Hassan and Arafain for telling me when it's time to go home. But no night could end nor sleep subsist without my ma or sisters wishing me goodnight. With the support f...

show abstract

Elastica as a dynamical system

Abstract: Abstract. The elastica is a curve in R 3 that is stationary under variations of the integral of the square of the curvature. Elastica are viewed as a dynamical system that arises from the second order calculus of variations, and its quantization is discussed.

Cited by 2 publications

References 13 publications

Generalizing parametrization invariance in the calculus of variations

Generalizing parametrization invariance in the calculus of variations

Geometric Recursive Dynamic Modelling and Simulation of Soft Robotic Systems

Contact Info

Product

Resources

About