Classical Dynamics of Fullerenes

Sławianowski, Jan J.; Kotowski, Romuald

doi:10.1007/s00033-017-0799-3

Cited by 3 publications

(5 citation statements)

References 51 publications

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“…We have seen that in classical mechanics, the geodetic affinely invariant models on

SL false(n, double-struckR false)

may describe elastic vibrations (see, eg, previous studies). Moreover, there exists a sharp threshold between finite vibrations and infinite escaping motions.…”

Section: Special Case Of Two Dimensionsmentioning

confidence: 91%

“…Later on, we frequently use also the so‐called 2‐polar decomposition of the configuration matrix, ie, φ = L D R −1 , where D is a diagonal matrix with elements on the diagonal representing the system of fundamental stretching

Q^{a} = \exp false(q^{a} false)

, and L , R are orthogonal matrices that represent the systems of eigenvectors of the Cauchy and Green deformation tensors normalized:

C L_{a} = λ_{a}^{- 1} L_{a} = \exp false(- 2 q^{a} false) L_{a}

G R_{a} = λ_{a} R_{a} = \exp false(2 q^{a} false) R_{a}

. When the spectrum is nondegenerate, then L a , R a are uniquely defined (up to reordering) and pair‐wise orthogonal:

η false(R_{a}, R_{b} false) = η_{A B} {R^{A}}_{a} {R^{B}}_{b} = δ_{a b} = g_{i j} {L^{i}}_{a} {L^{j}}_{b} = g false(L_{a}, L_{b} false) .

For more detailed description of the classical dynamics of the affinely rigid bodies see, eg, these studies …”

Section: Some Basic Concepts About Classical Mechanics Of Affinely Rimentioning

confidence: 99%

“…But, as said above, quite independently of those fundamental problems, almost every mechanical and field‐theoretical scheme admits those 2 formulations: classical and quantum. For example, large molecules or fullerenes may be described in a satisfactory way in both frameworks, every one to be used in its specific domain of applications (see, for instance, our most recent paper about classical dynamics of fullerenes and some earlier papers about quantized mechanics of collective and internal affine modes). Moreover, such micro‐ and nano‐objects are physically placed somewhere in the convolution region of 2 theories.…”

Section: General Scheme Of the Schrödinger Wave Mechanicsmentioning

confidence: 99%

“…Let us begin with a short review of our earlier results concerning the mechanics of affinely rigid bodies, ie, bodies, which deformative behaviour is restricted to performing homogeneous deformations only . They are an obvious affine counterpart of the usual metrically rigid bodies defined with respect to some Euclidean metric.…”

Section: Introductionmentioning

confidence: 99%

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Quantized mechanics of affinely rigid bodies

Sławianowski

Kovalchuk

Gołubowska

et al. 2017

Math Methods in App Sciences

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In this paper, we develop the main ideas of the quantized version of affinely rigid (homogeneously deformable) motion. We base our consideration on the usual Schrödinger formulation of quantum mechanics in the configuration manifold, which is given, in our case, by the affine group or equivalently by the semi-direct product of the linear group GL(n, R) and the space of translations R n , where n equals the dimension of the "physical space." In particular, we discuss the problem of dynamical invariance of the kinetic energy under the action of the whole affine group, not only under the isometry subgroup. Technically, the treatment is based on the 2-polar decomposition of the matrix of the internal configuration and on the Peter-Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of elastic media and microstructured continua. KEYWORDS homogeneously deformable body, Peter-Weyl analysis, Schrödinger quantization 6900

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“…We have seen that in classical mechanics, the geodetic affinely invariant models on

SL false(n, double-struckR false)

may describe elastic vibrations (see, eg, previous studies). Moreover, there exists a sharp threshold between finite vibrations and infinite escaping motions.…”

Section: Special Case Of Two Dimensionsmentioning

confidence: 91%

Q^{a} = \exp false(q^{a} false)

, and L , R are orthogonal matrices that represent the systems of eigenvectors of the Cauchy and Green deformation tensors normalized:

C L_{a} = λ_{a}^{- 1} L_{a} = \exp false(- 2 q^{a} false) L_{a}

G R_{a} = λ_{a} R_{a} = \exp false(2 q^{a} false) R_{a}

. When the spectrum is nondegenerate, then L a , R a are uniquely defined (up to reordering) and pair‐wise orthogonal:

η false(R_{a}, R_{b} false) = η_{A B} {R^{A}}_{a} {R^{B}}_{b} = δ_{a b} = g_{i j} {L^{i}}_{a} {L^{j}}_{b} = g false(L_{a}, L_{b} false) .

For more detailed description of the classical dynamics of the affinely rigid bodies see, eg, these studies …”

Section: Some Basic Concepts About Classical Mechanics Of Affinely Rimentioning

confidence: 99%

Section: General Scheme Of the Schrödinger Wave Mechanicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantized mechanics of affinely rigid bodies

Sławianowski

Kovalchuk

Gołubowska

et al. 2017

Math Methods in App Sciences

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“…In this article, we study the effect of the simplest deformation of a fullerite crystalline fragment on the dynamic characteristics of the C 60 molecule using the methods of classical mechanics [33][34][35][36][37]. A computational analysis of the behavior of the fullerene molecule depending on the speed, direction, and magnitude of the indentation deformation has been carried out.…”

Section: Introductionmentioning

confidence: 99%

Computational Analysis of Strain-Induced Effects on the Dynamic Properties of C60 in Fullerite

et al. 2022

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A hybrid discrete-continuous physical and mathematical model is used to study what deformation characteristics cause the rolling effect of C60 fullerene in a fullerite crystal. The interaction of fullerene atoms with surrounding molecules is described using a centrally symmetric interaction potential, in which the surrounding molecules are considered as a spherical surface of uniformly distributed carbon atoms. The rotational motion of fullerene is described by the Euler dynamic equations. The results of a numerical study of the influence of the rate, magnitude, and direction of strain on the dynamic characteristics of the rotational and translational motion of C60 fullerene in a crystalline fragment are presented.

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Numerical Study of the Passage of Natural Gas Components through C60 Fullerite in the Low-Temperature Phase

et al. 2022

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The movement of natural gas components through a crystalline fragment of C60 fullerite in a simple cubic phase is numerically investigated. The methods of classical molecular physics, the Monte Carlo and Euler approaches, and a step-by-step numerical method of a high order of accuracy are used to study the effects arising from the passage of the particles through the C60 fullerite. The influence of the characteristics of incoming particles on the permeability of fullerite was analyzed using the results of the performed calculations. The application of the fast integral approach based on the smoothed spherical potential is discussed and compared with the Monte Carlo method.

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Classical Dynamics of Fullerenes

Cited by 3 publications

References 51 publications

Quantized mechanics of affinely rigid bodies

Quantized mechanics of affinely rigid bodies

Computational Analysis of Strain-Induced Effects on the Dynamic Properties of C60 in Fullerite

Numerical Study of the Passage of Natural Gas Components through C60 Fullerite in the Low-Temperature Phase

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