Holonomic basis

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

\mathbf {e} _{\alpha }=\lim _{\delta x^{\alpha }\to 0}{\frac {\delta \mathbf {s} }{\delta x^{\alpha }}},

where $s$ is the infinitesimal displacement vector between the point $P$ and a nearby point $Q$ whose coordinate separation from $P$ is $δx α$ along the coordinate curve $x α$ (i.e. the curve on the manifold through $P$ for which the local coordinate $x α$ 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 $x α (λ)$ with the tangent vector $u = u α e α$ , where $u α = dx α / dλ$ , and a function $f (x α)$ defined in a neighbourhood of $C$ , the variation of $f$ along $C$ can be written as

{\frac {df}{d\lambda }}={\frac {dx^{\alpha }}{d\lambda }}{\frac {\partial f}{\partial x^{\alpha }}}=u^{\alpha }{\frac {\partial f}{\partial x^{\alpha }}}.

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

A local condition for a basis ${e k}$ to be holonomic is that all mutual Lie derivatives vanish:^[3]

\left[\mathbf {e} _{\alpha },\mathbf {e} _{\beta }\right]={\cal {L}}_{\mathbf {e} _{\alpha }}\mathbf {e} _{\beta }=0.

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

It is in general not possible to find a coordinate basis that is orthonomal in any open region $U$ of a manifold $M$ , with an obvious exception of the real coordinate space $R n$ considered as a manifold with the Euclidean metric $δ ij e i \otimes e j$ at every point.^[4]

References

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

Holonomic basis

References

See also