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The essential point is that their geometric assumptions, via some of the results discussed below on harmonic radius, give good control over harmonic coordinates on regions near infinity. By the use of a partition of unity, these harmonic coordinates can be patched together to form a single coordinate chart, which is the main objective. [19]
The harmonic coordinate condition is one of several coordinate conditions in general relativity, which make it possible to solve the Einstein field equations.A coordinate system is said to satisfy the harmonic coordinate condition if each of the coordinate functions x α (regarded as scalar fields) satisfies d'Alembert's equation.
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point , it is one of the most important model systems in quantum mechanics.
Unlike the harmonic and synchronous coordinate conditions, some commonly used coordinate conditions may be either under-determinative or over-determinative. An example of an under-determinative condition is the algebraic statement that the determinant of the metric tensor is −1, which still leaves considerable gauge freedom. [ 4 ]
The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion.The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics.
This gravitational model is complete to spherical harmonic degree and order 2159 (block diagonal), and contains additional coefficients extending to degree 2190 and order 2159. It provides a raster of 2.5′×2.5′ and an accuracy approaching 10 cm. 1'×1' is also available [ 7 ] in non-float but lossless PGM , [ 5 ] [ 8 ] but original .gsb ...
Within the context of a model system in classical mechanics, the phase-space coordinates of the system at any given time are composed of all of the system's dynamic variables. Because of this, it is possible to calculate the state of the system at any given time in the future or the past, through integration of Hamilton's or Lagrange's ...
Eells and Sampson introduced the harmonic map heat flow and proved the following fundamental properties: Regularity. Any harmonic map heat flow is smooth as a map (a, b) × M → N given by (t, p) ↦ f t (p). Now suppose that M is a closed manifold and (N, h) is geodesically complete. Existence.