Characterisation of cylindrical curves

Starostin, E. L.; Heijden, G. H. M. van der

doi:10.1007/s00605-014-0705-4

Cited by 11 publications

(19 citation statements)

References 4 publications

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“…Once the world lines of spinning particles are cylindrical, they comprise the class of interest in this paper. This class is much less studied in geometry, though there are some works on cylindrical lines in d = 3 Euclidean space 34,35 . In Minkowski space, the issue of general cylindric lines has not been studied yet, to the best of our knowledge.…”

Section: Equations Of Motion For World Lines On the World Sheetsmentioning

confidence: 99%

“…The differential equation for the lines has been obtained in an implicit form. In 35 , another implicit form of the cylindric line equations has been deduced employing an alternative ad hoc method in 3d Euclidean space. To the best of our knowledge, the problem of cylindric lines has never been solved in Minkowski space, even in d = 3.…”

Section: Equations Of Motion For World Lines On the World Sheetsmentioning

confidence: 99%

“…With two vector parameters subject to two scalar constraints, the variety of cylinders is a 4-parameter set of surfaces in d = 3 Minkowski space. To express these parameters in terms of derivatives of the line, one has to consider differential consequences of the algebraic equation (35) up to fourth order. Equations (30) for the cylinder are specified as…”

Section: A Lightlike World Lines On a Cylinder With A Timelike Axismentioning

confidence: 99%

“…Throughout this section, prime denotes derivative by τ . Differentiating the cylinder equation (35) four times and accounting for identities (53), we arrive at the relations connecting the constants n, y with the derivatives of the cylindric line (x ′ , x + n(n, x) − y)= 0 , (x ′′ , x + n(n, x) − y) + (n, x ′ ) 2 = 0 , (x ′′′ , x + n(n, x) − y) + 3(n, x ′′ )(n, x ′ )= 0 , (x ′′′′ , x + n(n, x) − y) + 4(n, x ′′′ )(n, x ′ ) + 3(n, x ′′ ) 2 − (x ′′ ) 2 = 0 .…”

Section: B Timelike World Lines On the Cylinder With Timelike Axismentioning

confidence: 99%

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World sheets of spinning particles

Kaparulin

Lyakhovich

2017

Phys. Rev. D

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The classical spinning particles are considered such that quantization of classical model leads to an irreducible massive representation of the Poincaré group. The class of gauge equivalent classical particle world lines is shown to form a [(d + 1)/2]-dimensional world sheet in d-dimensional Minkowski space, irrespectively to any specifics of the classical model. For massive spinning particles in d = 3, 4, the world sheets are shown to be circular cylinders. The radius of cylinder is fixed by representation. In higher dimensions, particle's world sheet turns out to be a toroidal cylinderProceeding from the fact that the world lines of irreducible classical spinning particles are cylindrical curves, while all the lines are gauge equivalent on the same world sheet, we suggest a method to deduce the classical equations of motion for particles and also to find their gauge symmetries. In d = 3 Minkowski space, the spinning particle path is defined by a single fourth-order differential equation having two zero-order gauge symmetries. The equation defines particle's path in Minkowski space, and it does not involve auxiliary variables. A special case is also considered of cylindric null curves, which are defined by a different system of equations. It is shown that the cylindric null curves also correspond to irreducible massive spinning particles. For the higher-derivative equation of motion of the irreducible massive spinning particle, we deduce the equivalent second-order formulation involving an auxiliary variable. The second-order formulation agrees with a previously known spinning particle model.

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Section: Equations Of Motion For World Lines On the World Sheetsmentioning

confidence: 99%

Section: Equations Of Motion For World Lines On the World Sheetsmentioning

confidence: 99%

Section: A Lightlike World Lines On a Cylinder With A Timelike Axismentioning

confidence: 99%

Section: B Timelike World Lines On the Cylinder With Timelike Axismentioning

confidence: 99%

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World sheets of spinning particles

Kaparulin

Lyakhovich

2017

Phys. Rev. D

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“…For differential geometry of the curves in (non-)Euclidean space we refer to the books [12,13]. The method of classification of cylindrical curves in Euclidean space has been proposed in [14,15]. We use this method with small modifications accounting the signature of the space-time metric.…”

Section: Introductionmentioning

confidence: 99%

Geometrical model of massive spinning particle in four-dimensional Minkowski space

Kaparulin

Lyakhovich

Retuntsev

2019

J. Phys.: Conf. Ser.

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We propose the model of massive spinning particle traveling in four-dimensional Minkowski space. The equations of motion of the particle follow from the fact that all the classical paths of the particle lie on a cylinder whose position in Minkowski space is determined by the particle's linear momentum and total angular momentum. All the paths on one and the same cylinder are gauge equivalent. The equations of motion are found in implicit form for general time-like paths, and they are non-Lagrangian. The explicit equations of motion are found for trajectories with small curvature and helices. The momentum and total angular momentum are expressed in terms of characteristics of the path in all the cases. The constructed model of the spinning particle has geometrical character, with no additional variables in the space of spin states being introduced.

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Moving frames and the characterization of curves that lie on a surface

Silva

Luiz²

2017

J. Geom.

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In this work we are interested in the characterization of curves that belong to a given surface. To the best of our knowledge, there is no known general solution to this problem. Indeed, a solution is only available for a few examples: planes, spheres, or cylinders. Generally, the characterization of such curves, both in Euclidean (E 3 ) and in Lorentz-Minkowski (E 3 1 ) spaces, involves an ODE relating curvature and torsion. However, by equipping a curve with a relatively parallel moving frame, Bishop was able to characterize spherical curves in E 3 through a linear equation relating the coefficients which dictate the frame motion. Here we apply these ideas to surfaces that are implicitly defined by a smooth function, Σ = F −1 (c), by reinterpreting the problem in the context of the metric given by the Hessian of F , which is not always positive definite. So, we are naturally led to the study of curves in E 3 1 . We develop a systematic approach to the construction of Bishop frames by exploiting the structure of the normal planes induced by the casual character of the curve, present a complete characterization of spherical curves in E 3 1 , and apply it to characterize curves that belong to a non-degenerate Euclidean quadric. We also interpret the casual character that a curve may assume when we pass from E 3 to E 3 1 and finally establish a criterion for a curve to lie on a level surface of a smooth function, which reduces to a linear equation when the Hessian is constant.

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Characterisation of cylindrical curves

Abstract: We employ moving frames along pairs of curves at constant separation to derive various conditions for a curve to belong to the surface of a circular cylinder.

Cited by 11 publications

References 4 publications

World sheets of spinning particles

World sheets of spinning particles

Geometrical model of massive spinning particle in four-dimensional Minkowski space

Moving frames and the characterization of curves that lie on a surface

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