Geometric description of time-dependent finite-dimensional mechanical systems

2020 ◽  
Vol 25 (11) ◽  
pp. 2050-2075
Author(s):  
Simon R. Eugster ◽  
Giuseppe Capobianco ◽  
Tom Winandy
Keyword(s):  
State Space ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Mechanical Systems ◽  
Second Order ◽  
The State ◽  
Time Dependent ◽  
Finite Dimensional ◽  
Affine Bundle ◽  
Time Position

Using the non-standard geometric structure proposed by Loos, we present a coordinate-free formulation of the theory for time-dependent finite-dimensional mechanical systems with n degrees of freedom. The state space containing the system’s information on time, position and velocity is defined as a (2 n+1)-dimensional affine bundle over an ( n+1)-dimensional generalized space-time. The main goal is to present a geometric postulate that characterizes a second-order vector field whose integral curves describe the motions of a time-dependent finite-dimensional mechanical system. The core objects of the postulate are differential two-forms on the state space, called action forms, which are in a bijective relation with second-order vector fields. The requirements for a differential two-form to be an action form allow for a coordinate-free definition of non-potential forces, which may depend on time, position and velocity. Finally, we show that not only Lagrange’s equations but also Hamilton’s equations follow directly as mere coordinate representations of the same coordinate-free postulate.

scholarly journals Characteristic phenomena in combustion

1997 ◽  
Vol 1 (2) ◽  
pp. 147-159
Author(s):  
Dirk Meinköhn
Keyword(s):  
State Space ◽  
Reaction Diffusion ◽  
Stationary Solutions ◽  
The State ◽  
State Variable ◽  
Finite Dimensional ◽  
Dimensional State Space ◽  
Singularity Order ◽  
The Relationship ◽  
Stable Stationary Solutions

For the case of a reaction–diffusion system, the stationary states may be represented by means of a state surface in a finite-dimensional state space. In the simplest example of a single semi-linear model equation given. in terms of a Fredholm operator, and under the assumption of a centre of symmetry, the state space is spanned by a single state variable and a number of independent control parameters, whereby the singularities in the set of stationary solutions are necessarily of the cuspoid type. Certain singularities among them represent critical states in that they form the boundaries of sheets of regular stable stationary solutions. Critical solutions provide ignition and extinction criteria, and thus are of particular physical interest. It is shown how a surface may be derived which is below the state surface at any location in state space. Its contours comprise singularities which correspond to similar singularities in the contours of the state surface, i.e., which are of the same singularity order. The relationship between corresponding singularities is in terms of lower bounds with respect to a certain distinguished control parameter associated with the name of Frank-Kamenetzkii.

A state-dependent lifetime process of individuals subject to external perturbations

1980 ◽  
Vol 17 (04) ◽  
pp. 922-938
Author(s):  
Wolfgang Mergenthaler
Keyword(s):  
Ionizing Radiation ◽  
State Space ◽  
Lifetime Distribution ◽  
The State ◽  
Time Dependent ◽  
Biological Cell ◽  
Regeneration Time ◽  
External Perturbations ◽  
State Dependent ◽  
The Mean

We consider an individual which ultimately dies or divides, and whose state is subject to drift and jumps caused by external perturbations. The mortality and division rates being state-dependent, the present paper deals with the time-dependent distribution of the individual's position in the state-space and with its lifetime distribution. The results are applied to a model of a biological cell which is exposed to ionizing radiation. Under certain conditions on the parameters of the type of perturbation one can show that the division probability decreases and the mean regeneration time increases with increasing frequency and ‘effect' of the perturbations.

scholarly journals SOME FEATURES OF THE STATE-SPACE TRAJECTORIES FOLLOWED BY ROBUST ENTANGLED FOUR-QUBIT STATES DURING DECOHERENCE

2010 ◽  
Vol 08 (03) ◽  
pp. 505-515 ◽  
Author(s):  
A. P. MAJTEY ◽  
A. BORRAS ◽  
A. R. PLASTINO ◽  
M. CASAS ◽  
A. PLASTINO
Keyword(s):  
State Space ◽  
Recent Work ◽  
Mixed State ◽  
Pure State ◽  
The State ◽  
Time Dependent ◽  
Space Trajectories ◽  
Geometrical Features ◽  
Initial States ◽  
Qubit States

In a recent work (Borras et al., Phys. Rev. A79 (2009) 022108), we have determined, for various decoherence channels, four-qubit initial states exhibiting the most robust possible entanglement. Here, we explore some geometrical features of the trajectories in state space generated by the decoherence process, connecting the initially robust pure state with the completely decohered mixed state obtained at the end of the evolution. We characterize these trajectories by recourse to the distance between the concomitant time-dependent mixed state and different reference states.

scholarly journals CONFORMAL FIELDS: A CLASS OF REPRESENTATIONS OF Vect(N)

1992 ◽  
Vol 07 (26) ◽  
pp. 6493-6508 ◽  
Author(s):  
T.A. LARSSON
Keyword(s):  
Differential Geometry ◽  
Representation Theory ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Tensor Fields ◽  
Normal Ordering ◽  
New Class ◽  
Finite Dimensional ◽  
Conformal Fields

Vect (N), the algebra of vector fields in N dimensions, is studied. Some aspects of local differential geometry are formulated as Vect(N) representation theory. There is a new class of modules, conformal fields, whose restrictions to the subalgebra sl(N+1)⊂ Vect (N) are finite-dimensional sl (N+1) representations. In this regard they are simpler than tensor fields. Fock modules are also constructed. Infinities, which are unremovable even by normal ordering, arise unless bosonic and fermionic degrees of freedom match.

Transient Vibration Analysis of Open Circular Cylindrical Shells

2005 ◽  
Vol 128 (3) ◽  
pp. 366-374 ◽  
Author(s):  
Selvakumar Kandasamy ◽  
Anand V. Singh
Keyword(s):  
State Space ◽  
Degrees Of Freedom ◽  
Grid Point ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Middle Surface ◽  
The State ◽  
Displacement Fields ◽  
Transient Vibration ◽  
Rotational Components

A numerical method based on the Rayleigh-Ritz method has been presented for the forced vibration of open cylindrical shells. The equations are derived from the three-dimensional strain-displacement relations in the cylindrical coordinate system. The middle surface of the shell represents the geometry, which is defined by an angle that subtends the curved edges, the length, and the thickness. The displacement fields are generated with a predefined set of grid points on the middle surface using considerably high-order polynomials. Each grid point has five degrees of freedom, viz., three translational components along the cylindrical coordinates and two rotational components of the normal to the middle surface. Then the strain and kinetic energy expressions are obtained in terms of these displacement fields. The differential equation governing the vibration characteristics of the shell is expressed in terms of the mass, stiffness, and the load consistent with the prescribed displacement fields. The transient response of the shell with and without damping is sought by transforming the equation of motion to the state-space model and then the state-space differential equations are solved using the Runge-Kutta algorithm.

scholarly journals Existence and non-existence for time-dependent mean field games with strong aggregation

2021 ◽  
Author(s):  
Marco Cirant ◽  
Daria Ghilli
Keyword(s):  
Population Density ◽  
State Space ◽  
Time Horizon ◽  
Existence Of Solutions ◽  
Mean Field ◽  
The State ◽  
Time Dependent ◽  
Classical Solutions ◽  
Mean Field Games ◽  
Quadratic Mean

AbstractWe investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings of the form $$-\sigma m^\alpha $$ - σ m α ,$$\alpha \ge 2/N$$ α ≥ 2 / N , where m is the population density and N is the dimension of the state space. We prove the existence of solutions under the assumption that $$\sigma $$ σ is small enough. For large $$\sigma $$ σ , we show that existence may fail whenever the time horizon T is large.

scholarly journals ADAPTIVE WAVELETS SLIDING MODE CONTROL FOR A CLASS OF SECOND ORDER UNDERACTUATED MECHANICAL SYSTEMS

2021 ◽  
Vol 61 (2) ◽  
pp. 350-363
Author(s):  
Fares Nafa ◽  
Aimad Boudouda ◽  
Billel Smaani
Keyword(s):  
Sliding Mode Control ◽  
Degrees Of Freedom ◽  
Sliding Mode ◽  
Mechanical Systems ◽  
Adaptive Controller ◽  
Descent Method ◽  
Second Order ◽  
Gradient Descent Method ◽  
Underactuated Mechanical Systems ◽  
Mode Control

The control of underactuated mechanical systems (UMS) remains an attracting field where researchers can develop their control algorithms. To this date, various linear and nonlinear control techniques using classical and intelligent methods have been published in literature. In this work, an adaptive controller using sliding mode control (SMC) and wavelets network (WN) is proposed for a class of second-order UMS with two degrees of freedom (DOF).This adaptive control strategy takes advantage of both sliding mode control and wavelet properties. In the main result, we consider the case of un-modeled dynamics of the above-mentioned UMS, and we introduce a wavelets network to design an adaptive controller based on the SMC. The update algorithms are directly extracted by using the gradient descent method and conditions are then settled to achieve the required convergence performance.The efficacy of the proposed adaptive approach is demonstrated through an application to the pendubot.

Response and Discretization Methods for Axially Moving Materials

1991 ◽  
Vol 44 (11S) ◽  
pp. S279-S284 ◽  
Author(s):  
J. A. Wickert ◽  
C. D. Mote
Keyword(s):  
State Space ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
The State ◽  
Space Formulation ◽  
Axially Moving ◽  
Effective Use ◽  
Gyroscopic Systems ◽  
Acceleration Component

Through a convective acceleration component, the equations of motion for axially-moving materials are skew-symmetric in the state space formulation, so that the response problem is best analyzed within the broader context of continuous gyroscopic systems. With particular application to the prototypical traveling string and beam models, a modal analysis that associates degrees of freedom with the complex state eigenfunctions and their conjugates is presented. This procedure is well-suited for harmonic excitation sources, and in some instances, it is more convenient than previous methods which decompose the modal coordinates, eigenfunctions, and generalized forces into real and imaginary components. Also from the state space perspective, Rayleigh’s quotient for gyroscopic systems provides a variational method for determining the eigensolutions of axially-moving materials. Ritz discretization of the quotient can make effective use of the speed-adapting modes of the traveling string and beam models as they are rich in phase, as well as amplitude, content.

scholarly journals Determinizing Asynchronous Automata on Infinite Inputs

1995 ◽  
Vol 2 (58) ◽  
Author(s):  
Nils Klarlund ◽  
Madhavan Mukund ◽  
Milind Sohoni
Keyword(s):  
State Space ◽  
Blow Up ◽  
Decision Procedures ◽  
Second Order ◽  
The State ◽  
Regular Languages ◽  
Logical Connective ◽  
Machine Model ◽  
Asynchronous Automata ◽  
Acceptance Condition

Asynchronous automata are a natural distributed machine model<br />for recognizing trace languages - languages defined over an alphabet<br />equipped with an independence relation.<br />To handle infinite traces, Gastin and Petit introduced Buchi asynchronous<br />automata, which accept precisely the class of omega-regular trace<br />languages. Like their sequential counterparts, these automata need to<br />be non-deterministic in order to capture all omega-regular languages. Thus<br />complementation of these automata is non-trivial. Complementation<br />is an important operation because it is fundamental for treating the<br />logical connective "not" in decision procedures for monadic second-order<br />logics. Subsequently, Diekert and Muscholl solved the complementation<br />problem by showing that with a Muller acceptance condition, deterministic<br />automata suffice for recognizing omega-regular trace languages.<br />However, a direct determinization procedure, extending the classical<br />subset construction, has proved elusive.<br />In this paper, we present a direct determinization procedure for<br />Buchi asynchronous automata, which generalizes Safra's construction<br />for sequential Buchi automata. As in the sequential case, the blow-up<br />in the state space is essentially that of the underlying subset construction.

scholarly journals Jet methods in time-dependent Lagrangian biomechanics

Open Physics ◽  
2010 ◽  
Vol 8 (5) ◽  
Author(s):  
Tijana Ivancevic
Keyword(s):  
Human Body ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Differential Forms ◽  
Flow Equation ◽  
Body Motion ◽  
Time Dependent ◽  
Metric Tensor ◽  
Geometric Evolution ◽  
Configuration Manifold

AbstractIn this paper we propose the time-dependent generalization of an ‘ordinary’ autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds associated to the extended musculo-skeletal configuration manifold, called the configuration bundle. We start with an ordinary configuration manifold of human body motion, given as a set of its all active degrees of freedom (DOF) for a particular movement. This is a Riemannian manifold with a material metric tensor given by the total mass-inertia matrix of the human body segments. This is the base manifold for standard autonomous biomechanics. To make its time-dependent generalization, we need to extend it with a real time axis. By this extension, using techniques from fibre bundles, we defined the biomechanical configuration bundle. On the biomechanical bundle we define vector-fields, differential forms and affine connections, as well as the associated jet manifolds. Using the formalism of jet manifolds of velocities and accelerations, we develop the time-dependent Lagrangian biomechanics. Its underlying geometric evolution is given by the Ricci flow equation.

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