Holonomic basis

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In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold M is a set of basis vector fields {eα} defined at every point P of a region of the manifold as

where s is the infinitesimal displacement vector between the point P and a nearby point Q whose coordinate separation from P is δxα along the coordinate curve xα (i.e. the curve on the manifold through P for which the coordinate xα varies and all other coordinates are constant).[1]

It is possible to make an association between such a basis and directional derivative operators. Given a parameterized curve C on the manifold defined by xα(λ) with the tangent vector u = uαeα, where uα = dxα/, and a function f(xα) defined in a neighbourhood of C, the variation of f along C can be written as

Since we have that u = uαeα, the identification is often made between a coordinate basis vector eα and the partial derivative operator /xα, under the interpretation of all vector relations as equalities between operators acting on scalar quantities.[2]

A local condition for a basis {ek} to be holonomic is (with this interpretation) that all mutual Lie derivatives vanish:[3]

A basis that is not holonomic is called a non-holonomic or non-coordinate basis.

It is generally impossible to find a coordinate basis that is also orthonomal in any open region U of a manifold M, with an obvious exception of the real coordinate space Rn considered as a manifold with the Euclidean metric δij at every point.[4]

References[edit]

  1. ^ M. P. Hobson; G. P. Efstathiou; A. N. Lasenby (2006), General Relativity: An Introduction for Physicists, Cambridge University Press, p. 57 
  2. ^ T. Padmanabhan (2010), Gravitation: Foundations and Frontiers, Cambridge University Press, p. 25 
  3. ^ Roger Penrose; Wolfgang Rindler, Spinors and Space–Time: Volume 1, Two-Spinor Calculus and Relativistic Fields, Cambridge University Press, pp. 197–199 
  4. ^ Bernard F. Schutz (1980), Geometrical Methods of Mathematical Physics, Cambridge University Press, p. 69, ISBN 9780521298872 

See also[edit]