Cartans Magic FormulaEdit
Cartan's magic formula is a central result in differential geometry that expresses how a differential form changes along the flow generated by a vector field. Named after Élie Cartan, the formula links three fundamental operations on forms: the Lie derivative, the exterior derivative, and the interior product. It holds on any smooth manifold M for any vector field X and any differential form on M, and it underpins a wide range of computations in geometry, physics, and beyond.
Intuitively, Cartan's formula provides a compact expression for the infinitesimal change of a form along a direction given by X. It reveals that the Lie derivative L_X is the graded commutator of the exterior derivative d and the interior product i_X, namely L_X = i_X d + d i_X. Through this identity, one can reduce the often complicated task of differentiating along flows to two more elementary operations.
Statement and interpretation
Let M be a smooth manifold, X a vector field on M, and ω a differential form on M. The Cartan's formula, also called Cartan's magic formula, states: L_X ω = i_X dω + d(i_X ω)
- Here, i_X ω denotes the interior product of X with ω, yielding a form of degree one less than ω.
- dω is the exterior derivative of ω, raising its degree by one.
- L_X ω is the Lie derivative of ω along X, describing how ω changes when pulled back along the flow generated by X.
This identity is a graded version of a product rule: applying the flow generated by X to a form can be decomposed into two pieces, one from the variation of the form itself (dω) and one from the variation of the argument along X (i_X ω). It also clarifies that L_X is a derivation of degree zero with respect to the wedge product, satisfying a Leibniz-type rule.
In coordinates, the formula can be verified by comparing how L_X acts on functions, one-forms, and higher-degree forms. For a function f, for instance, L_X f = X(f) and i_X f = 0, so L_X f = i_X df. For a one-form α, the identity becomes L_X α = i_X dα + d(i_X α), which matches the standard coordinate expression for the Lie derivative of a one-form.
Historical background
Élie Cartan developed the language of exterior calculus and moving frames in the early 20th century, and Cartan's formula emerges as a natural consequence of his framework. The identity is now a staple in the toolkit of differential geometry, informing not only pure mathematics but also the mathematical formulations that appear in physics and gauge theory. Cartan’s work on symmetries and differential forms laid groundwork that later influenced fields as diverse as symplectic geometry and gauge theory.
Examples and consequences
- Functions (0-forms): If f is a function, i_X f = 0, so L_X f = i_X df. This recovers the familiar notion that the Lie derivative of a function along X equals the directional derivative of f in the direction X.
- 1-forms: For a 1-form α, L_X α = i_X dα + d(i_X α). This is a standard tool for checking whether a form is preserved under a flow.
- Closed forms and Hamiltonian dynamics: If ω is a closed form (dω = 0), then L_X ω = d(i_X ω). In particular, for a symplectic form ω (where dω = 0), L_X ω = d(i_X ω). A vector field X with L_X ω = 0 preserves the symplectic structure, a condition central to the study of Hamiltonian dynamics and symplectic geometry.
- Killing fields and metric structures: When ω is constructed from a metric (for example, as the metric tensor or associated volume form), Cartan's formula relates infinitesimal symmetries of the geometric structure to the behavior of forms under X. If X generates an isometry, L_X g = 0 for the metric g.
Applications and connections
- Differential geometry and topology: Cartan's formula is a workhorse for manipulating Lie derivatives of forms in proofs and constructions, such as in the study of invariants, foliations, and de Rham cohomology.
- Symplectic geometry and classical mechanics: The identity clarifies how flows generated by vector fields interact with the symplectic form and with observables encoded as differential forms. It underpins the notion of Hamiltonian vector fields, where i_X ω is often the differential of a Hamiltonian function.
- General relativity and gauge theories: The Lie derivative encodes how fields transform under diffeomorphisms; Cartan's formula gives a practical way to compute these variations for differential forms that appear in field equations and conservation laws.