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 {e₁, ..., e_n} defined at every point P of a region of the manifold as

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

where δs 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^αe_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

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

Since we have that u = u^αe_α, the identification is often made between a coordinate basis vector e_α and the partial derivative operator ⁠∂/∂x^α⁠, under the interpretation of vectors as operators acting on functions.^[2]

A local condition for a basis {e₁, ..., e_n} to be holonomic is that all mutual Lie derivatives vanish:^[3]

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

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 Rⁿ considered as a manifold with g being the Euclidean metric δ_ijeⁱ ⊗ e^j at every point.