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A Variational Formulation for Relativistic Mechanics based on Riemannian Geometry and its Application to the Quantum Mechanics Context
Published in Fabio Silva Botelho, Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering, 2020
Remark 30.1 If the connection in question is such that Γjki=12gil(∂gkl∂uj+∂gjl∂uk−∂gjk∂ul) such a connection is said to be the Levi-Civita one. In the next lines we assume the concerning connection is indeed the Levi-Civita one.
The Calculus of Variations
Published in George F. Simmons, Differential Equations with Applications and Historical Notes, 2016
In this example we were able to solve equations (28) quite easily, but in general this task is extremely difficult. The main significance of these equations lies in their connection with the following very important result in mathematical physics: if a particle glides along a surface, free from the action of any external force, then its path is a geodesic. We shall prove this dynamical theorem in Appendix B. For the purpose of this argument it will be convenient to assume that the parameter t is the arc length s measured along the curve, so that f = 1 and equations (28) become () d2x/ds2Gx=d2y/ds2Gy=d2z/ds2Gz.
Parallel transport of local strains
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2019
F. Milicchio, V. Varano, S. Gabriele, L. Teresi, P. E. Puddu, P. Piras
Biological applications usually analyze shapes and forms utilizing homologous anatomical landmarks, and applying the discretization to a continuum model. Shapes are embedded in curved Riemannian manifolds, and to achieve an interpolation of shapes one needs to discover the geodesic underlying the deformation paths. Riemannian manifolds are defined by a metric and a compatible connection, as detailed in do Carmo Valero (1992). Both properties permit analyses on manifolds by allowing the computations of distances between points, i.e. metrical properties, and moving on the manifold along paths called geodesics. The Parallel Transport (PT) is a connection that allows transporting geometrical data along geodesics defined on manifolds, and with a choice of connection compatible with the metric, given two vectors at a point, geometrical properties such as the angle in between and their length will be conserved after applying a PT.