Lie bracket of vector fields

In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted [X, Y].

Conceptually, the Lie bracket [X, Y] is the derivative of Y along the flow generated by X, and is sometimes denoted ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by X.

The Lie bracket is an R-bilinear operation and turns the set of all smooth vector fields on the manifold M into an (infinite-dimensional) Lie algebra.

The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental in the geometric theory of nonlinear control systems.[1]

DefinitionsEdit

There are three conceptually different but equivalent approaches to defining the Lie bracket:

Vector fields as derivationsEdit

Each smooth vector field X on a manifold M may be regarded as a differential operator acting on smooth functions C(M). Indeed, each smooth vector field X becomes a derivation on C(M) when we define X(f) to be a function whose value at a point p is the directional derivative of f at p in the direction X(p). Furthermore, any derivation on C(M) arises from a unique smooth vector field X.

In general, the commutator   of any two derivations   and   is again a derivation, where   denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:

 

Flows and limitsEdit

Let   be the flow associated with the vector field X, and let D denote the tangent map derivative operator. Then the Lie bracket of X and Y at the point xM can be defined as the Lie derivative:

 

This also measures the failure of the flow in the successive directions   to return to the point x:

 

In coordinatesEdit

Though the above definitions of Lie bracket are intrinsic (independent of the choice of coordinates on the manifold M), in practice one often wants to compute the bracket in terms of a specific coordinate system  . We write   for the associated local basis of the tangent bundle, so that general vector fields can be written  and  for smooth functions  . Then the Lie bracket can be computed as:

 

If M is (an open subset of) Rn, then the vector fields X and Y can be written as smooth maps of the form   and  , and the Lie bracket   is given by:

 

where   and   are n×n Jacobian matrices multiplying the 1 column vectors X and Y.

PropertiesEdit

The Lie bracket of vector fields equips the real vector space   of all vector fields on M (i.e., smooth sections of the tangent bundle  ) with the structure of a Lie algebra, which means [ • , • ] is a map   with:

  • R-bilinearity
  • Anti-symmetry,  
  • Jacobi identity,  

An immediate consequence of the second property is that   for any  .

Furthermore, there is a "product rule" for Lie brackets. Given a smooth (scalar-valued) function f on M and a vector field Y on M, we get a new vector field fY by multiplying the vector Yx by the scalar f(x) at each point xM. Then:

  •  

where we multiply the scalar function X(f) with the vector field Y, and the scalar function f with the vector field [X, Y]. This turns the vector fields with the Lie bracket into a Lie algebroid.

Vanishing of the Lie bracket of X and Y means that following the flows in these directions defines a surface embedded in M, with X and Y as coordinate vector fields:

Theorem:   iff the flows of X and Y commute locally, meaning   for all xM and sufficiently small s, t.

This is a special case of the Frobenius integrability theorem.

ExamplesEdit

For a Lie group G, the corresponding Lie algebra   is the tangent space at the identity  , which can be identified with the vector space of left invariant vector fields on G. The Lie bracket of two left invariant vector fields is also left invariant, which defines the Jacobi–Lie bracket operation  .

For a matrix Lie group, whose elements are matrices  , each tangent space can be represented as matrices:  , where   means matrix multiplication and I is the identity matrix. The invariant vector field corresponding to  is given by  , and a computation shows the Lie bracket on   corresponds to the usual commutator of matrices:

 

ApplicationsEdit

The Jacobi–Lie bracket is essential to proving small-time local controllability (STLC) for driftless affine control systems.

GeneralizationsEdit

As mentioned above, the Lie derivative can be seen as a generalization of the Lie bracket. Another generalization of the Lie bracket (to vector-valued differential forms) is the Frölicher–Nijenhuis bracket.

ReferencesEdit

  • Hazewinkel, Michiel, ed. (2001) [1994], "Lie bracket", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Isaiah, Pantelis (2009), "Controlled parking [Ask the experts]", IEEE Control Systems Magazine, 29 (3): 17–21, 132, doi:10.1109/MCS.2009.932394
  • Khalil, H.K. (2002), Nonlinear Systems (3rd ed.), Upper Saddle River, NJ: Prentice Hall, ISBN 0-13-067389-7
  • Kolář, I., Michor, P., and Slovák, J. (1993), Natural operations in differential geometry, Springer-VerlagCS1 maint: multiple names: authors list (link) Extensive discussion of Lie brackets, and the general theory of Lie derivatives.
  • Lang, S. (1995), Differential and Riemannian manifolds, Springer-Verlag, ISBN 978-0-387-94338-1 For generalizations to infinite dimensions.
  • Lewis, Andrew D., Notes on (Nonlinear) Control Theory (PDF)[permanent dead link]
  • Warner, Frank (1983) [1971], Foundations of differentiable manifolds and Lie groups, New York-Berlin: Springer-Verlag, ISBN 0-387-90894-3