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Normal coordinates exist on a normal neighborhood of a point p in M. A normal neighborhood U is an open subset of M such that there is a proper neighborhood V of the origin in the tangent space T p M, and exp p acts as a diffeomorphism between U and V. On a normal neighborhood U of p in M, the chart is given by: :=: The isomorphism E, and ...
The normal coordinates, denoted as Q, refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time.
In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the line perpendicular to the tangent line to the curve at the point. A normal vector of length one is called a unit normal vector.
This space-dependence is called a normal mode. Usually, for problems with continuous dependence on (x, y, z) there is no single or finite number of normal modes, but there are infinitely many normal modes. If the problem is bounded (i.e. it is defined on a finite section of space) there are countably many normal modes (usually numbered n = 1, 2 ...
The GF method, sometimes referred to as FG method, is a classical mechanical method introduced by Edgar Bright Wilson to obtain certain internal coordinates for a vibrating semi-rigid molecule, the so-called normal coordinates Q k. Normal coordinates decouple the classical vibrational motions of the molecule and thus give an easy route to ...
An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations = and = (but these may not hold at other points in the frame). These coordinates are also called normal coordinates.
For each point, there exist coordinate systems in which the Christoffel symbols vanish at the point. [16] These are called (geodesic) normal coordinates, and are often used in Riemannian geometry. There are some interesting properties which can be derived directly from the transformation law.
If itself is a geodesic, then Fermi–Walker's transport becomes the standard parallel transport and Fermi's coordinates become standard Riemannian coordinates adapted to . In this case, using these coordinates in a neighbourhood T {\displaystyle T} of γ {\displaystyle \gamma } , we have Γ b c a = 0 {\displaystyle \Gamma _{bc}^{a}=0} , all ...