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 ${e 1, ..., e n}$ 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 $δ s$ is the 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 $uα = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}dxα/dλ$ , 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 }{\partial x^{\alpha }}}f.

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 vectors as operators acting on scalar quantities.^[2]

A local condition for a basis ${e 1, ..., e n}$ to be holonomic is that all mutual Lie derivatives vanish:^[3]

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

A basis that is not holonomic is called an anholonomic,^[4] non-holonomic or non-coordinate basis.

Given a metric tensor $g$ on a manifold $M$ , it is in general not possible to find a coordinate basis that is orthonormal in any open region $U$ of $M$ .^[5] An obvious exception is when $M$ is the real coordinate space $R n$ considered as a manifold with $g$ being the Euclidean metric $δ ij e i \otimes e j$ at every point.

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
^ Charles W. Misner, Kip S. Thorne, John Archibald Wheeler (1970), Gravitation, p. 210CS1 maint: uses authors parameter (link)
^ Bernard F. Schutz (1980), Geometrical Methods of Mathematical Physics, Cambridge University Press, pp. 47–49, ISBN 9780521298872

Holonomic basis

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