10:45 AM - 12:15 PM
[PEM10-P19] Development of Affine Connection for Describing Spatiotemporal Evolving 3D Magnetic Vector Fields and its applications
Keywords:Affine Connection, Spatiotemporal Evolving 3D Magnetic Vector Fields
We have developed an affine connection on a differentiable manifold for analyzing and formulating the geometrical structure of magnetic fields as vector fields evolving arbitrarily in space-time.
A manifold is a space that can define a local Euclidean space at all points belonging to it. At each point on the manifold, a tangent space (tangent vector space) is defined. The tangent vector space is the set of all tangent vectors passing through the points of contact, forming a tangent vector field. This tangent vector field can be assumed to be a variety of calculus laws established in Euclidean space in the vicinity of the points of contact on the manifold. By connecting the tangent vector space defined at each of these points with the surrounding tangent vector space by a parallel shift on the manifold, the vector field in the entire space on the manifold can be defined. This connection of tangent vector spaces by parallel shifts on the manifold is an affine connection. That is, the affine connection allows the tangent vector field defined at each point on the manifold to behave like a function with values in the vector space fixed at each point. In other words, a collection of local coordinate systems defined at each point on the manifold covers the entire space, and data and physics can be described and compared under a common rule.
The affine connection we have developed enables us to capture the vector field as a fiber bundle of tightly distributed mag- netic field lines and to describe its geometrical structure in the framework of ordinary vector analysis. This method extends the locally defined orthonormal Cartesian coordinate system parallel and perpendicular to the magnetic field in 3D by the following methodology:
(1) . To investigate the geometry of magnetic field lines, we first introduce the Frenet frame, which is a dynamic frame along a single curved line. In the Frenet frame, the local principal curve direction is determined by the unit vector in the tangent direction of the curve of interest, and the unit vectors in the principal normal (curvature direction) and secondary normal (orthogonal to the above two) directions to the principal curve are used to form a ”local normal orthogonal system ” is formed. Considering a tightly distributed group of magnetic field lines, this local normal orthogonal system will be distributed in space.
(2) . To connect the local orthonormal orthogonal systems distributed in this space, vector curved lines formed by connecting unit vectors in the main normal and bi-normal directions are introduced, and by shifting along each curve, the relative relationship with adjacent magnetic field lines is grasped and the analytical space is extended to the entire system.
(3). (1) and (2) uniquely determine the distribution of curvature and twist ratio for each curve filling the space tightly. This determines how the ”tetrahedron” formed by the local orthonormal basis changes along with each curve (rigid directional change and rotation), and the 3-dimensional geometric structure of the magnetic field is completely determined.
That is, by shifting the tetrahedron formed by the local coordinate system along the principal curve, the principal normal, and the bi-normal, respectively, we can understand how the moving tetrahedron curved and rotated along the curved line, thereby determining the structure of the vector field that extends over the entire space. This is a methodology to know the structure of the vector field expanding in the whole space. Furthermore, the temporal evolution ratio of this tetrahedron (temporal changing ratio of direction and twisting ratio) is derived. This allows us to describe the 3-dimensional evolution of the magnetic field geometry.
In the talk, we will give an overview of the fundamental structure of the vector field obtained by these frameworks and discuss the underlying physics-based determinants of the vector field and its applications.
A manifold is a space that can define a local Euclidean space at all points belonging to it. At each point on the manifold, a tangent space (tangent vector space) is defined. The tangent vector space is the set of all tangent vectors passing through the points of contact, forming a tangent vector field. This tangent vector field can be assumed to be a variety of calculus laws established in Euclidean space in the vicinity of the points of contact on the manifold. By connecting the tangent vector space defined at each of these points with the surrounding tangent vector space by a parallel shift on the manifold, the vector field in the entire space on the manifold can be defined. This connection of tangent vector spaces by parallel shifts on the manifold is an affine connection. That is, the affine connection allows the tangent vector field defined at each point on the manifold to behave like a function with values in the vector space fixed at each point. In other words, a collection of local coordinate systems defined at each point on the manifold covers the entire space, and data and physics can be described and compared under a common rule.
The affine connection we have developed enables us to capture the vector field as a fiber bundle of tightly distributed mag- netic field lines and to describe its geometrical structure in the framework of ordinary vector analysis. This method extends the locally defined orthonormal Cartesian coordinate system parallel and perpendicular to the magnetic field in 3D by the following methodology:
(1) . To investigate the geometry of magnetic field lines, we first introduce the Frenet frame, which is a dynamic frame along a single curved line. In the Frenet frame, the local principal curve direction is determined by the unit vector in the tangent direction of the curve of interest, and the unit vectors in the principal normal (curvature direction) and secondary normal (orthogonal to the above two) directions to the principal curve are used to form a ”local normal orthogonal system ” is formed. Considering a tightly distributed group of magnetic field lines, this local normal orthogonal system will be distributed in space.
(2) . To connect the local orthonormal orthogonal systems distributed in this space, vector curved lines formed by connecting unit vectors in the main normal and bi-normal directions are introduced, and by shifting along each curve, the relative relationship with adjacent magnetic field lines is grasped and the analytical space is extended to the entire system.
(3). (1) and (2) uniquely determine the distribution of curvature and twist ratio for each curve filling the space tightly. This determines how the ”tetrahedron” formed by the local orthonormal basis changes along with each curve (rigid directional change and rotation), and the 3-dimensional geometric structure of the magnetic field is completely determined.
That is, by shifting the tetrahedron formed by the local coordinate system along the principal curve, the principal normal, and the bi-normal, respectively, we can understand how the moving tetrahedron curved and rotated along the curved line, thereby determining the structure of the vector field that extends over the entire space. This is a methodology to know the structure of the vector field expanding in the whole space. Furthermore, the temporal evolution ratio of this tetrahedron (temporal changing ratio of direction and twisting ratio) is derived. This allows us to describe the 3-dimensional evolution of the magnetic field geometry.
In the talk, we will give an overview of the fundamental structure of the vector field obtained by these frameworks and discuss the underlying physics-based determinants of the vector field and its applications.