Information AboutGeodesics |
| CATEGORIES ABOUT GEODESIC | |
| fundamental physics concepts | |
| riemannian geometry | |
| metric geometry | |
| hamiltonian mechanics | |
|
Definition of geodesic depends on the type of "curved space". If the space carries a natural Metric then geodesics are defined to be ( Locally ) the shortest path between points on the space. The term "geodesic" comes from '' Geodesy '', the science of measuring the size and shape of the Earth ; in the original sense, a geodesic was the shortest route between two points on the Surface of the earth, namely, a Segment of a Great Circle . INTRODUCTION The shortest path between two points in a curved space can be found by writing the Equation for the length of a Curve , and then minimizing this length using standard techniques of Calculus and Differential Equations . Equivalently, a different quantity may be defined, termed the Energy of the curve; minimizing the energy leads to the same equations for a geodesic. Intuitively, one can understand this second formulation by noting that an Elastic Band stretched between two points will contract its length, and in so doing will minimize its energy; the resulting shape of the band is a geodesic. Geodesics are commonly seen in the study of Riemannian Geometry and more generally Metric Geometry . In Physics , geodesics describe the motion of Point Particle s; in particular, the path taken by a falling rock, an orbiting Satellite , or the shape of a Planetary Orbit are all described by geodesics in the theory of General Relativity . More generally, the topic of Sub-Riemannian Geometry deals with the paths that objects may take when they are not free, and their movement is constrained in various ways. This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian and Pseudo-Riemannian Manifold s. The article Geodesic (general Relativity) discusses the special case of general relativity in greater detail. Examples The most familiar examples are the straight lines in Euclidean Geometry . On a Sphere , the geodesics are the Great Circle s. The shortest path from point ''A'' to point ''B'' on a sphere is given by the shorter piece of the great circle passing through ''A'' and ''B''. If ''A'' and ''B'' are Antipodal Point s (like the North pole and the South pole), then there are infinitely ''many'' shortest paths between them. METRIC GEOMETRY In ''I'' to the Metric Space ''M'' is a geodesic if there is a Constant ''v'' ≥ 0 such that for any ''t'' ∈ ''I'' there is a neighborhood ''J'' of ''t'' in ''I'' such that for any ''t''1, ''t''2 ∈ ''J'' we have | ||
| :<math>d(\gamma(t 1),\gamma(t 2)) | t_1-t_2\,</math> |
|
|