Modifcations to gravitational waves due to matter shells
- Authors: Naidoo, Monogaran
- Date: 2021-10-29
- Subjects: Gravitational waves , General relativity (Physics) , Einstein field equations , Cosmology , Matter shells
- Language: English
- Type: Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/191118 , vital:45062 , 10.21504/10962/191119
- Description: As detections of gravitational waves (GWs) mount, the need to investigate various effects on the propagation of these waves from the time of emission until detection also grows. We investigate how a thin low density dust shell surrounding a gravitational wave source affects the propagation of GWs. The Bondi-Sachs (BS) formalism for the Einstein equations is used for the problem of a gravitational wave (GW) source surrounded by a spherical dust shell. Using linearised perturbation theory, we and the geometry of the regions exterior to, interior to and within the shell. We and that the dust shell causes the gravitational wave to be modified both in magnitude and phase, but without any energy being transferred to or from the dust. This finding is novel. In the context of cosmology, apart from the gravitational redshift, the effects are too small to be measurable; but the effect would be measurable if a GW event were to occur with a source surrounded by a massive shell and with the radius of the shell and the wavelength of the GWs of the same order. We extended our investigation to astrophysical scenarios such as binary black hole (BBH) mergers, binary neutron star (BNS) mergers, and core collapse supernovae (CCSNe). In these scenarios, instead of a monochromatic GW source, as we used in our initial investigation, we consider burst-like GW sources. The thin density shell approach is modified to include thick shells by considering concentric thin shells and integrating. Solutions are then found for these burst-like GW sources using Fourier transforms. We show that GW echoes that are claimed to be present in the Laser Interferometer Gravitational-Wave Observatory (LIGO) data of certain events, could not have been caused by a matter shell. We do and, however, that matter shells surrounding BBH mergers, BNS mergers, and CCSNe could make modifications of order a few percent to a GW signal. These modifications are expected to be measurable in GW data with current detectors if the event is close enough and at a detectable frequency; or in future detectors with increased frequency range and amplitude sensitivity. Substantial use is made of computer algebra in these investigations. In setting the scene for our investigations, we trace the evolution of general relativity (GR) from Einstein's postulation in 1915 to vindication of his theory with the confirmation of the existence of GWs a century later. We discuss the implications of our results to current and future considerations. Calculations of GWs, both analytical and numerical, have normally assumed their propagation from source to a detector on Earth in a vacuum spacetime, and so discounted the effect of intervening matter. As we enter an era of precision GW measurements, it becomes important to quantify any effects due to propagation of GWs through a non-vacuum spacetime Observational confirmation of the modification effect that we and in astrophysical scenarios involving black holes (BHs), neutron stars (NSs) and CCSNe, would also enhance our understanding of the details of the physics of these bodies. , Thesis (PhD) -- Faculty of Science, Mathematics (Pure and Applied), 2021
- Full Text:
- Authors: Naidoo, Monogaran
- Date: 2021-10-29
- Subjects: Gravitational waves , General relativity (Physics) , Einstein field equations , Cosmology , Matter shells
- Language: English
- Type: Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/191118 , vital:45062 , 10.21504/10962/191119
- Description: As detections of gravitational waves (GWs) mount, the need to investigate various effects on the propagation of these waves from the time of emission until detection also grows. We investigate how a thin low density dust shell surrounding a gravitational wave source affects the propagation of GWs. The Bondi-Sachs (BS) formalism for the Einstein equations is used for the problem of a gravitational wave (GW) source surrounded by a spherical dust shell. Using linearised perturbation theory, we and the geometry of the regions exterior to, interior to and within the shell. We and that the dust shell causes the gravitational wave to be modified both in magnitude and phase, but without any energy being transferred to or from the dust. This finding is novel. In the context of cosmology, apart from the gravitational redshift, the effects are too small to be measurable; but the effect would be measurable if a GW event were to occur with a source surrounded by a massive shell and with the radius of the shell and the wavelength of the GWs of the same order. We extended our investigation to astrophysical scenarios such as binary black hole (BBH) mergers, binary neutron star (BNS) mergers, and core collapse supernovae (CCSNe). In these scenarios, instead of a monochromatic GW source, as we used in our initial investigation, we consider burst-like GW sources. The thin density shell approach is modified to include thick shells by considering concentric thin shells and integrating. Solutions are then found for these burst-like GW sources using Fourier transforms. We show that GW echoes that are claimed to be present in the Laser Interferometer Gravitational-Wave Observatory (LIGO) data of certain events, could not have been caused by a matter shell. We do and, however, that matter shells surrounding BBH mergers, BNS mergers, and CCSNe could make modifications of order a few percent to a GW signal. These modifications are expected to be measurable in GW data with current detectors if the event is close enough and at a detectable frequency; or in future detectors with increased frequency range and amplitude sensitivity. Substantial use is made of computer algebra in these investigations. In setting the scene for our investigations, we trace the evolution of general relativity (GR) from Einstein's postulation in 1915 to vindication of his theory with the confirmation of the existence of GWs a century later. We discuss the implications of our results to current and future considerations. Calculations of GWs, both analytical and numerical, have normally assumed their propagation from source to a detector on Earth in a vacuum spacetime, and so discounted the effect of intervening matter. As we enter an era of precision GW measurements, it becomes important to quantify any effects due to propagation of GWs through a non-vacuum spacetime Observational confirmation of the modification effect that we and in astrophysical scenarios involving black holes (BHs), neutron stars (NSs) and CCSNe, would also enhance our understanding of the details of the physics of these bodies. , Thesis (PhD) -- Faculty of Science, Mathematics (Pure and Applied), 2021
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Best simultaneous approximation in normed linear spaces
- Authors: Johnson, Solomon Nathan
- Date: 2018
- Subjects: Normed linear spaces , Approximation theory , Mathematical analysis
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/58985 , vital:27400
- Description: In this thesis we consider the problem of simultaneously approximating elements of a set B C X by a single element of a set K C X. This type of a problem arises when the element to be approximated is not known precisely but is known to belong to a set.Thus, best simultaneous approximation is a natural generalization of best approximation which has been studied extensively. The theory of best simultaneous approximation has been studied by many authors, see for example [4], [8], [25], [28], [26] and [12] to name but a few.
- Full Text:
- Authors: Johnson, Solomon Nathan
- Date: 2018
- Subjects: Normed linear spaces , Approximation theory , Mathematical analysis
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/58985 , vital:27400
- Description: In this thesis we consider the problem of simultaneously approximating elements of a set B C X by a single element of a set K C X. This type of a problem arises when the element to be approximated is not known precisely but is known to belong to a set.Thus, best simultaneous approximation is a natural generalization of best approximation which has been studied extensively. The theory of best simultaneous approximation has been studied by many authors, see for example [4], [8], [25], [28], [26] and [12] to name but a few.
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Geometry of deformed special relativity
- Authors: Sixaba, Vuyile
- Date: 2018
- Subjects: Special relativity (Physics) , Quantum gravity , Quantum theory , Geometry , Heisenberg uncertainty principle
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/59478 , vital:27615
- Description: We undertake a study of the classical regime in which Planck's constant and Newton's gravitational constant are negligible, but not their ratio, the Planck mass, in hopes that this could possibly lead to testable quantum gravity (QG) effects in a classical regime. In this quest for QG phenomenology we consider modifications of the standard dispersion relation of a free particle known as deformed special relativity (DSR). We try to geometrize DSR to find the geometric origin of the spacetime and momentum space. In particular, we adopt the framework of Hamilton geometry which is set up on phase space, as the cotangent bundle of configuration space in order to derive a purely phase space formulation of DSR. This is necessary when one wants to understand potential links of DSR with modifications of quantum mechanics such as Generalised Uncertainty Principles. It is subsequently observed that space-time and momentum space emerge naturally as curved and intertwined spaces. In conclusion we mention examples and applications of this framework as well as potential future developments.
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- Authors: Sixaba, Vuyile
- Date: 2018
- Subjects: Special relativity (Physics) , Quantum gravity , Quantum theory , Geometry , Heisenberg uncertainty principle
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/59478 , vital:27615
- Description: We undertake a study of the classical regime in which Planck's constant and Newton's gravitational constant are negligible, but not their ratio, the Planck mass, in hopes that this could possibly lead to testable quantum gravity (QG) effects in a classical regime. In this quest for QG phenomenology we consider modifications of the standard dispersion relation of a free particle known as deformed special relativity (DSR). We try to geometrize DSR to find the geometric origin of the spacetime and momentum space. In particular, we adopt the framework of Hamilton geometry which is set up on phase space, as the cotangent bundle of configuration space in order to derive a purely phase space formulation of DSR. This is necessary when one wants to understand potential links of DSR with modifications of quantum mechanics such as Generalised Uncertainty Principles. It is subsequently observed that space-time and momentum space emerge naturally as curved and intertwined spaces. In conclusion we mention examples and applications of this framework as well as potential future developments.
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Contributions to the study of nonholonomic Riemannian manifolds
- Authors: Barrett, Dennis Ian
- Date: 2017
- Subjects: Riemannian manifolds , Curvature , Lie groups , Geometry, Riemannian , Tensor fields
- Language: English
- Type: Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/7554 , vital:21272 , DOI https://doi.org/10.21504/10962/7554
- Description: In this thesis we consider nonholonomic Riemannian manifolds, and in particular, left- invariant nonholonomic Riemannian structures on Lie groups. These structures are closely related to mechanical systems with (positive definite) quadratic Lagrangians and nonholo- nomic constraints linear in velocities. In the first chapter, we review basic concepts of non- holonomic Riemannian geometry, including the left-invariant structures. We also examine the class of left-invariant structures with so-called Cartan-Schouten connections. The second chapter investigates the curvature of nonholonomic Riemannian manifolds and the Schouten and Wagner curvature tensors. The Schouten tensor is canonically associated to every non- holonomic Riemannian structure (in particular, we use it to define isometric invariants for structures on three-dimensional manifolds). By contrast, the Wagner tensor is not generally intrinsic, but can be used to characterise flat structures (i.e., those whose associated parallel transport is path-independent). The third chapter considers equivalence of nonholonomic Rie- mannian manifolds, particularly up to nonholonomic isometry. We also introduce the notion of a nonholonomic Riemannian submanifold, and investigate the conditions under which such a submanifold inherits its geometry from the enveloping space. The latter problem involves the concept of a geodesically invariant distribution, and we show it is also related to the curvature. In the last chapter we specialise to three-dimensional nonholonomic Riemannian manifolds. We consider the equivalence of such structures up to nonholonomic isometry and rescaling, and classify the left-invariant structures on the (three-dimensional) simply connected Lie groups. We also characterise the flat structures in three dimensions, and then classify the flat structures on the simply connected Lie groups. Lastly, we consider three typical examples of (left-invariant) nonholonomic Riemannian structures on three-dimensional Lie groups, two of which arise from problems in classical mechanics (viz., the Chaplygin problem and the Suslov problem). , Thesis (PhD) -- Faculty of Science, Mathematics (Pure and Applied), 2017
- Full Text:
- Authors: Barrett, Dennis Ian
- Date: 2017
- Subjects: Riemannian manifolds , Curvature , Lie groups , Geometry, Riemannian , Tensor fields
- Language: English
- Type: Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/7554 , vital:21272 , DOI https://doi.org/10.21504/10962/7554
- Description: In this thesis we consider nonholonomic Riemannian manifolds, and in particular, left- invariant nonholonomic Riemannian structures on Lie groups. These structures are closely related to mechanical systems with (positive definite) quadratic Lagrangians and nonholo- nomic constraints linear in velocities. In the first chapter, we review basic concepts of non- holonomic Riemannian geometry, including the left-invariant structures. We also examine the class of left-invariant structures with so-called Cartan-Schouten connections. The second chapter investigates the curvature of nonholonomic Riemannian manifolds and the Schouten and Wagner curvature tensors. The Schouten tensor is canonically associated to every non- holonomic Riemannian structure (in particular, we use it to define isometric invariants for structures on three-dimensional manifolds). By contrast, the Wagner tensor is not generally intrinsic, but can be used to characterise flat structures (i.e., those whose associated parallel transport is path-independent). The third chapter considers equivalence of nonholonomic Rie- mannian manifolds, particularly up to nonholonomic isometry. We also introduce the notion of a nonholonomic Riemannian submanifold, and investigate the conditions under which such a submanifold inherits its geometry from the enveloping space. The latter problem involves the concept of a geodesically invariant distribution, and we show it is also related to the curvature. In the last chapter we specialise to three-dimensional nonholonomic Riemannian manifolds. We consider the equivalence of such structures up to nonholonomic isometry and rescaling, and classify the left-invariant structures on the (three-dimensional) simply connected Lie groups. We also characterise the flat structures in three dimensions, and then classify the flat structures on the simply connected Lie groups. Lastly, we consider three typical examples of (left-invariant) nonholonomic Riemannian structures on three-dimensional Lie groups, two of which arise from problems in classical mechanics (viz., the Chaplygin problem and the Suslov problem). , Thesis (PhD) -- Faculty of Science, Mathematics (Pure and Applied), 2017
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Contributions to the study of a class of optimal control problems on the orthogonal groups SO(3) and SO(4)
- Authors: Adams, Ross Montague
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64826 , vital:28608
- Description: In this thesis we investigate a class of invariant optimal control problems, and their associated quadratic Hamilton-Poisson systems, on the orthogonal groups SO(3) and SO(4). Specifically, we are concerned with the class of left-invariant control affine systems. We begin by classifying all cost-extended systems on SO(3) under cost equivalence. (Cost-extended systems are closely related to optimal control problems.) A classification of all quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space so(3)*, under affine equivalence, is also obtained. For the normal forms obtained in our classification (of Hamilton-Poisson systems) we investigate the (Lyapunov) stability nature of the equilibria using spectral and energy-Casimir methods. For a subclass of these systems, we obtain analytic expressions for the integral curves of the associated Hamiltonian vector fields in terms of (basic) Jacobi elliptic functions. The explicit relationship between the classification of cost-extended systems on SO(3) and the classification of quadratic Hamilton- Poisson systems on so(3)* is provided. On SO(4), a classification of all left-invariant control affine systems under L-equivalence is obtained. We then determine which of these representatives are controllable, thus obtaining a classification under detached feedback equivalence. We also obtain a partial classification of quadratic Hamilton-Poisson systems on the Lie-Poisson space so(4)*. An investigation of the stability nature of the equilibria for a subclass of these systems is also done. Several illustrative examples of optimal control problems on the orthogonal group SO(3) are provided. More specifically, we consider an optimal control problem corresponding to a representative of our classification (of cost-extended system) for each possible number of control inputs. For each of these problems, we obtain explicit expressions for the extremal trajectories on the homogeneous space S2 by projecting the extremal trajectories on the group SO(3). The examples provided show how our classifications of cost-extended systems and Hamilton-Poisson systems can be used to obtain the optimal controls and the extremal trajectories corresponding to a large class of optimal control problems on SO(3). An example of a four-input optimal control problem on SO(4) is also provided. This example is provided to show how the solutions of certain problems on SO(4) can be related to the solutions of certain optimal control problems on SO(3).
- Full Text:
- Authors: Adams, Ross Montague
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64826 , vital:28608
- Description: In this thesis we investigate a class of invariant optimal control problems, and their associated quadratic Hamilton-Poisson systems, on the orthogonal groups SO(3) and SO(4). Specifically, we are concerned with the class of left-invariant control affine systems. We begin by classifying all cost-extended systems on SO(3) under cost equivalence. (Cost-extended systems are closely related to optimal control problems.) A classification of all quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space so(3)*, under affine equivalence, is also obtained. For the normal forms obtained in our classification (of Hamilton-Poisson systems) we investigate the (Lyapunov) stability nature of the equilibria using spectral and energy-Casimir methods. For a subclass of these systems, we obtain analytic expressions for the integral curves of the associated Hamiltonian vector fields in terms of (basic) Jacobi elliptic functions. The explicit relationship between the classification of cost-extended systems on SO(3) and the classification of quadratic Hamilton- Poisson systems on so(3)* is provided. On SO(4), a classification of all left-invariant control affine systems under L-equivalence is obtained. We then determine which of these representatives are controllable, thus obtaining a classification under detached feedback equivalence. We also obtain a partial classification of quadratic Hamilton-Poisson systems on the Lie-Poisson space so(4)*. An investigation of the stability nature of the equilibria for a subclass of these systems is also done. Several illustrative examples of optimal control problems on the orthogonal group SO(3) are provided. More specifically, we consider an optimal control problem corresponding to a representative of our classification (of cost-extended system) for each possible number of control inputs. For each of these problems, we obtain explicit expressions for the extremal trajectories on the homogeneous space S2 by projecting the extremal trajectories on the group SO(3). The examples provided show how our classifications of cost-extended systems and Hamilton-Poisson systems can be used to obtain the optimal controls and the extremal trajectories corresponding to a large class of optimal control problems on SO(3). An example of a four-input optimal control problem on SO(4) is also provided. This example is provided to show how the solutions of certain problems on SO(4) can be related to the solutions of certain optimal control problems on SO(3).
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Invariant control systems and sub-Riemannian structures on lie groups: equivalence and isometries
- Authors: Biggs, Rory
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64815 , vital:28607
- Description: In this thesis we consider invariant optimal control problems and invariant sub-Riemannian structures on Lie groups. Primarily, we are concerned with the equivalence and classification of problems (resp. structures). In the first chapter, both the class of invariant optimal control problems and the class of invariant sub-Riemannian structures are organised as categories. The latter category is shown to be functorially equivalent to a subcategory of the former category. Via the Pontryagin Maximum Principle, we associate to each invariant optimal control problem (resp. invariant sub-Riemannian structure) a quadratic Hamilton-Poisson system on the associated Lie-Poisson space. These Hamiltonian systems are also organised as a category and hence the Pontryagin lift is realised as a contravariant functor. Basic properties of these categories are investigated. The rest of this thesis is concerned with the classification (and investigation) of certain subclasses of structures and systems. In the second chapter, the homogeneous positive semidefinite quadratic Hamilton-Poisson systems on three-dimensional Lie-Poisson spaces are classified up to compatibility with a linear isomorphism; a list of 23 normal forms is exhibited. In the third chapter, we classify the invariant sub-Riemannian structures in three dimensions and calculate the isometry group for each normal form. By comparing our results with known results, we show that most isometries (in three dimensions) are in fact the composition of a left translation and a Lie group isomorphism. In the fourth and last chapter of this thesis, we classify the sub-Riemannian and Riemannian structures on the (2n + 1)-dimensional Heisenberg groups. Furthermore, we find the isometry group and geodesics of each normal form.
- Full Text:
- Authors: Biggs, Rory
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64815 , vital:28607
- Description: In this thesis we consider invariant optimal control problems and invariant sub-Riemannian structures on Lie groups. Primarily, we are concerned with the equivalence and classification of problems (resp. structures). In the first chapter, both the class of invariant optimal control problems and the class of invariant sub-Riemannian structures are organised as categories. The latter category is shown to be functorially equivalent to a subcategory of the former category. Via the Pontryagin Maximum Principle, we associate to each invariant optimal control problem (resp. invariant sub-Riemannian structure) a quadratic Hamilton-Poisson system on the associated Lie-Poisson space. These Hamiltonian systems are also organised as a category and hence the Pontryagin lift is realised as a contravariant functor. Basic properties of these categories are investigated. The rest of this thesis is concerned with the classification (and investigation) of certain subclasses of structures and systems. In the second chapter, the homogeneous positive semidefinite quadratic Hamilton-Poisson systems on three-dimensional Lie-Poisson spaces are classified up to compatibility with a linear isomorphism; a list of 23 normal forms is exhibited. In the third chapter, we classify the invariant sub-Riemannian structures in three dimensions and calculate the isometry group for each normal form. By comparing our results with known results, we show that most isometries (in three dimensions) are in fact the composition of a left translation and a Lie group isomorphism. In the fourth and last chapter of this thesis, we classify the sub-Riemannian and Riemannian structures on the (2n + 1)-dimensional Heisenberg groups. Furthermore, we find the isometry group and geodesics of each normal form.
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Invariant optimal control on the three-dimensional semi-Euclidean group: control affine and quadratic Hamilton-Poisson systems
- Authors: Barrett, Dennis Ian
- Date: 2014
- Subjects: Automorphisms , Symmetry (Mathematics) , Lyapunov stability , Geometry, Riemannian , Geometry, Affine , Elliptic functions
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/64805 , vital:28605
- Description: In this thesis we consider invariant control systems and Hamilton-Poisson systems on the three dimensional semi-Euclidean group SE(1,1). We first classify the left-invariant control affine systems (under detached feedback equivalence). We provide a complete list of normal forms, as well as classifying conditions. As a corollary to this classification, we derive controllability criteria for control affine systems on SE(1,1). Secondly, we consider quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space se(1,1)*. These systems are classified up to an affine isomorphism. Six normal forms are identified for the homogeneous case, whereas sixteen representatives (including several infinite families) are obtained for the inhomogeneous systems. Thereafter we consider the stability and integration of the normal forms obtained. For all homogeneous systems, and a subclass of inhomogeneous systems, we perform a complete stability analysis and derive explicit expressions for all integral curves. (The extremal controls of a large class of optimal control problems on SE(1,1) are linearly related to these integral curves.) Lastly, we discuss the Riemannian and sub-Riemannian problems. The (left-invariant) Riemannian and sub-Riemannian structures on SE(1,1) are classified, up to isometric group automorphisms and scaling. Explicit expressions for the geodesics of the normalised structures are found.
- Full Text:
- Authors: Barrett, Dennis Ian
- Date: 2014
- Subjects: Automorphisms , Symmetry (Mathematics) , Lyapunov stability , Geometry, Riemannian , Geometry, Affine , Elliptic functions
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/64805 , vital:28605
- Description: In this thesis we consider invariant control systems and Hamilton-Poisson systems on the three dimensional semi-Euclidean group SE(1,1). We first classify the left-invariant control affine systems (under detached feedback equivalence). We provide a complete list of normal forms, as well as classifying conditions. As a corollary to this classification, we derive controllability criteria for control affine systems on SE(1,1). Secondly, we consider quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space se(1,1)*. These systems are classified up to an affine isomorphism. Six normal forms are identified for the homogeneous case, whereas sixteen representatives (including several infinite families) are obtained for the inhomogeneous systems. Thereafter we consider the stability and integration of the normal forms obtained. For all homogeneous systems, and a subclass of inhomogeneous systems, we perform a complete stability analysis and derive explicit expressions for all integral curves. (The extremal controls of a large class of optimal control problems on SE(1,1) are linearly related to these integral curves.) Lastly, we discuss the Riemannian and sub-Riemannian problems. The (left-invariant) Riemannian and sub-Riemannian structures on SE(1,1) are classified, up to isometric group automorphisms and scaling. Explicit expressions for the geodesics of the normalised structures are found.
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