Applied Differential Geometry - A Modern Introduction

Ivancevic, Vladimir G.; Ivancevic, Tijana T.

doi:10.1142/9789812770721

Cited by 58 publications

(257 citation statements)

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“…Autonomous Hamiltonian biomechanics (as well as autonomous Lagrangian biomechanics), based on the postulate of conservation of the total mechanical energy, can be derived from the covariant force law [2][3][4][5], which in 'plain English' states:…”

Section: The Covariant Force Lawmentioning

confidence: 99%

“…We develop autonomous Hamiltonian biomechanics on the configuration biomechanical manifold M in three steps, following the standard symplectic geometry prescription (see [2,4,5,8]…”

Section: Autonomous Hamiltonian Biomechanicsmentioning

confidence: 99%

“…Most of dynamics in contemporary human biomechanics is autonomous (see [2][3][4][5][6][7][8][9][10][11]). This approach works fine for most individual movement simulations and predictions, in which the total human energy dissipations are insignificant.…”

Section: Introductionmentioning

confidence: 99%

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Jet spaces in modern Hamiltonian biomechanics

Ivancevic¹,

Jovanovic

Stanković

et al. 2010

Open Physics

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Abstract:In this paper we propose the time-dependent Hamiltonian form of human biomechanics, as a sequel to our previous work in time-dependent Lagrangian biomechanics [1]. This is the time-dependent generalization of an 'ordinary' autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. In our view, this time-dependent energetic approach is much more realistic than the autonomous one. Starting with the Covariant Force Law, we first develop autonomous Hamiltonian biomechanics. Then we extend it using a powerful geometrical machinery consisting of fibre bundles and jet manifolds associated to the biomechanical configuration manifold. We derive time-dependent, dissipative, Hamiltonian equations and the fitness evolution equation for the general time-dependent human biomechanical system.

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Section: The Covariant Force Lawmentioning

confidence: 99%

“…We develop autonomous Hamiltonian biomechanics on the configuration biomechanical manifold M in three steps, following the standard symplectic geometry prescription (see [2,4,5,8]…”

Section: Autonomous Hamiltonian Biomechanicsmentioning

confidence: 99%

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Jet spaces in modern Hamiltonian biomechanics

Ivancevic¹,

Jovanovic

Stanković

et al. 2010

Open Physics

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“…The simplest, mechanical-like LSF-action in the individual's LSF-manifold Σ has a Riemannian locomotion form (summation convention is always assumed) Dynamics of N DOF mechanical-like systems with action (14) and Hamiltonian (15) are commonly given by the set of geodesic equations (Ivancevic, 2006b;2007a) …”

Section: Geometric Chaos and Topological Phase Transitionsmentioning

confidence: 99%

“…In this geometrical framework, the instability of the trajectories is the instability of the geodesics, and it is completely determined by the curvature properties of the LSF-manifold Σ according to the Jacobi equation of geodesic deviation (see Ivancevic, 2006b;2007a) Using the Eisenhart metric (17), the relevant part of the Jacobi equation (18) is given by the tangent dynamics equation (Casetti et al, 1996;Caiani et al, 1997) 2 00 2 0,(1,...,),…”

Section: Geometric Chaos and Topological Phase Transitionsmentioning

confidence: 99%

Nonlinear Quantum Neuro-Psycho-Dynamics with Topological Phase Transitions

Ivancevic

2008

Neuroquantology

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We have proposed a novel model of general quantum, stochastic and chaotic psychodynamics. The model is based on the previously developed Life-Space Foam (LSF) framework to motivational and c ognitive dynamics. The present model extends the LSF-approach by incorporating chaotic and topological non-equilibrium phase transitions. Such extended LSF-model is applied for rigorous description of multi-agent joint action. The present model is related to Haken-Kelso-Bunz model of self-organization in the human motor system (including: multi-stability, phase transitions and hysteresis effects, presenting a contrary view to the purely feedback driven neural systems), as well as the entropy-approach to adaptation in human goal-directed motor control.

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A pseudo‐spectral method that uses an overlapping multidomain technique for the numerical solution of sine‐Gordon equation in one and two spatial dimensions

Taleei

Dehghan

2013

Math Methods in App Sciences

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In this article, we study an explicit scheme for the solution of sine-Gordon equation when the space discretization is carried out by an overlapping multidomain pseudo-spectral technique. By using differentiation matrices, the equation is reduced to a nonlinear system of ordinary differential equations in time that can be discretized with the explicit fourth-order Runge-Kutta method. To achieve approximation with high accuracy in large domains, the number of space grid points must be large enough. This yields very large and full matrices in the pseudo-spectral method that causes large memory requirements. The domain decomposition approach provides sparsity in the matrices obtained after the discretization, and this property reduces storage for large matrices and provides economical ways of performing matrix-vector multiplications. Therefore, we propose a multidomain pseudo-spectral method for the numerical simulation of the sineGordon equation in large domains. Test examples are given to demonstrate the accuracy and capability of the proposed method. Numerical experiments show that the multidomain scheme has an excellent long-time numerical behavior for the sine-Gordon equation in one and two dimensions.

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Applied Differential Geometry - A Modern Introduction

Cited by 58 publications

References 0 publications

Jet spaces in modern Hamiltonian biomechanics

Jet spaces in modern Hamiltonian biomechanics

Nonlinear Quantum Neuro-Psycho-Dynamics with Topological Phase Transitions

A pseudo‐spectral method that uses an overlapping multidomain technique for the numerical solution of sine‐Gordon equation in one and two spatial dimensions

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