Moving frame
In
Introduction
In lay terms, a
The Frenet–Serret frame plays a key role in the
In the late 19th century,
Later, moving frames were developed extensively by
- A ordered basis of a vector space.
- An orthogonal unit vectors (an orthonormal basis).
- An affine frame of an affine space consists of a choice of origin along with an ordered basis of vectors in the associated difference space.[4]
- A Euclidean frame of an affine space is a choice of origin along with an orthonormal basis of the difference space.
- A linearly independentpoints in the space.
- vierbeins, in German.
In each of these examples, the collection of all frames is homogeneous in a certain sense. In the case of linear frames, for instance, any two frames are related by an element of the general linear group. Projective frames are related by the projective linear group. This homogeneity, or symmetry, of the class of frames captures the geometrical features of the linear, affine, Euclidean, or projective landscape. A moving frame, in these circumstances, is just that: a frame which varies from point to point.
Formally, a frame on a homogeneous space G/H consists of a point in the tautological bundle G → G/H. A moving frame is a section of this bundle. It is moving in the sense that as the point of the base varies, the frame in the fibre changes by an element of the symmetry group G. A moving frame on a submanifold M of G/H is a section of the pullback of the tautological bundle to M. Intrinsically[5] a moving frame can be defined on a principal bundle P over a manifold. In this case, a moving frame is given by a G-equivariant mapping φ : P → G, thus framing the manifold by elements of the Lie group G.
One can extend the notion of frames to a more general case: one can "
Although there is a substantial formal difference between extrinsic and intrinsic moving frames, they are both alike in the sense that a moving frame is always given by a mapping into G. The strategy in Cartan's method of moving frames, as outlined briefly in
Method of the moving frame
Cartan (1937) formulated the general definition of a moving frame and the method of the moving frame, as elaborated by Weyl (1938). The elements of the theory are
- A Lie group G.
- A Klein spaceX whose group of geometric automorphisms is G.
- A smooth manifoldΣ which serves as a space of (generalized) coordinates for X.
- A collection of frames ƒ each of which determines a coordinate function from X to Σ (the precise nature of the frame is left vague in the general axiomatization).
The following axioms are then assumed to hold between these elements:
- There is a free and transitive group action of G on the collection of frames: it is a principal homogeneous spacefor G. In particular, for any pair of frames ƒ and ƒ′, there is a unique transition of frame (ƒ→ƒ′) in G determined by the requirement (ƒ→ƒ′)ƒ = ƒ′.
- Given a frame ƒ and a point A ∈ X, there is associated a point x = (A,ƒ) belonging to Σ. This mapping determined by the frame ƒ is a bijection from the points of X to those of Σ. This bijection is compatible with the law of composition of frames in the sense that the coordinate x′ of the point A in a different frame ƒ′ arises from (A,ƒ) by application of the transformation (ƒ→ƒ′). That is,
Of interest to the method are parameterized submanifolds of X. The considerations are largely local, so the parameter domain is taken to be an open subset of Rλ. Slightly different techniques apply depending on whether one is interested in the submanifold along with its parameterization, or the submanifold up to reparameterization.
Moving tangent frames
The most commonly encountered case of a moving frame is for the bundle of tangent frames (also called the frame bundle) of a manifold. In this case, a moving tangent frame on a manifold M consists of a collection of vector fields e1, e2, …, en forming a basis of the tangent space at each point of an open set U ⊂ M.
If is a coordinate system on U, then each vector field ej can be expressed as a linear combination of the coordinate vector fields :
Coframes
A moving frame determines a dual frame or coframe of the cotangent bundle over U, which is sometimes also called a moving frame. This is a n-tuple of smooth 1-forms
- θ1, θ2, …, θn
which are linearly independent at each point q in U. Conversely, given such a coframe, there is a unique moving frame e1, e2, …, en which is dual to it, i.e., satisfies the duality relation θi(ej) = δij, where δij is the Kronecker delta function on U.
If is a coordinate system on U, as in the preceding section, then each covector field θi can be expressed as a linear combination of the coordinate covector fields :
In the setting of classical mechanics, when working with canonical coordinates, the canonical coframe is given by the tautological one-form. Intuitively, it relates the velocities of a mechanical system (given by vector fields on the tangent bundle of the coordinates) to the corresponding momenta of the system (given by vector fields in the cotangent bundle; i.e. given by forms). The tautological one-form is a special case of the more general solder form, which provides a (co-)frame field on a general fiber bundle.
Uses
Moving frames are important in general relativity, where there is no privileged way of extending a choice of frame at an event p (a point in spacetime, which is a manifold of dimension four) to nearby points, and so a choice must be made. In contrast in special relativity, M is taken to be a vector space V (of dimension four). In that case a frame at a point p can be translated from p to any other point q in a well-defined way. Broadly speaking, a moving frame corresponds to an observer, and the distinguished frames in special relativity represent inertial observers.
In relativity and in
Further details
A moving frame always exists locally, i.e., in some neighbourhood U of any point p in M; however, the existence of a moving frame globally on M requires
on the Earth's surface break down as a moving frame at the north and south poles.The method of moving frames of Élie Cartan is based on taking a moving frame that is adapted to the particular problem being studied. For example, given a curve in space, the first three derivative vectors of the curve can in general define a frame at a point of it (cf. torsion tensor for a quantitative description – it is assumed here that the torsion is not zero). In fact, in the method of moving frames, one more often works with coframes rather than frames. More generally, moving frames may be viewed as sections of principal bundles over open sets U. The general Cartan method exploits this abstraction using the notion of a Cartan connection.
Atlases
In many cases, it is impossible to define a single frame of reference that is valid globally. To overcome this, frames are commonly pieced together to form an
Generalizations
Although this article constructs the frame fields as a coordinate system on the tangent bundle of a manifold, the general ideas move over easily to the concept of a vector bundle, which is a manifold endowed with a vector space at each point, that vector space being arbitrary, and not in general related to the tangent bundle.
Applications
Aircraft maneuvers can be expressed in terms of the moving frame (Aircraft principal axes) when described by the pilot.
See also
- Darboux frame
- Frenet–Serret formulas
- Yaw, pitch, and roll
Notes
- ^ a b Chern 1985
- ^ D. J. Struik, Lectures on classical differential geometry, p. 18
- ^ a b c Griffiths 1974
- ^ "Affine frame" Proofwiki.org
- ^ See Cartan (1983) 9.I; Appendix 2 (by Hermann) for the bundle of tangent frames. Fels and Olver (1998) for the case of more general fibrations. Griffiths (1974) for the case of frames on the tautological principal bundle of a homogeneous space.
References
- Cartan, Élie (1937), La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Paris: Gauthier-Villars.
- Cartan, Élie (1983), Geometry of Riemannian Spaces, Math Sci Press, Massachusetts.
- Chern, S.-S.(1985), "Moving frames", Elie Cartan et les Mathematiques d'Aujourd'hui, Asterisque, numero hors serie, Soc. Math. France, pp. 67–77.
- Cotton, Émile (1905), "Genéralisation de la theorie du trièdre mobile", Bull. Soc. Math. France, 33: 1–23.
- Darboux, Gaston (1887), Leçons sur la théorie génerale des surfaces, vol. I, Gauthier-Villars.
- Darboux, Gaston (1915), Leçons sur la théorie génerale des surfaces, vol. II, Gauthier-Villars.
- Darboux, Gaston (1894), Leçons sur la théorie génerale des surfaces, vol. III, Gauthier-Villars.
- Darboux, Gaston (1896), Leçons sur la théorie génerale des surfaces, vol. IV, Gauthier-Villars.
- Ehresmann, C. (1950), "Les connexions infinitésimals dans un espace fibré differential", Colloque de Topologie, Bruxelles, pp. 29–55.
- Evtushik, E.L. (2001) [1994], "Moving-frame method", Encyclopedia of Mathematics, EMS Press.
- Fels, M.; S2CID 826629.
- Green, M (1978), "The moving frame, differential invariants and rigidity theorem for curves in homogeneous spaces", S2CID 120620785.
- S2CID 12966544
- Guggenheimer, Heinrich (1977), Differential Geometry, New York: Dover Publications.
- Sharpe, R. W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Berlin, New York: ISBN 978-0-387-94732-7.
- Spivak, Michael (1999), A Comprehensive introduction to differential geometry, vol. 3, Houston, TX: Publish or Perish.
- Sternberg, Shlomo (1964), Lectures on Differential Geometry, Prentice Hall.
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