In calculus, the Lie derivative designated (after Sophus Lie ) is the derivative of a vector field, or more generally a tensor field along a vector field. With the help of a Lie bracket can be defined for vector fields, so that the set of all vector fields is a differentiable manifold to a Lie algebra.
In the general theory of relativity and the geometric formulation of Hamiltonian mechanics, the Lie derivative is used to detect symmetries, to exploit them to solve problems and find, for example, constants of the motion.
Lie derivative for functions
Is a vector field, the Lie derivative of a differentiable function of the application of:
There were an -dimensional manifold, a smooth function and a smooth vector field on. The Lie derivative of the function in accordance with the point is defined as the direction of discharge by:
In local coordinates, the vector field can be represented as
For the Lie derivative is then as follows
Lie derivative of vector fields
Are and two vector fields, then the Lie derivative of by the Lie bracket of the two vector fields:
This is defined as follows:
Be two smooth vector fields on a smooth manifold -dimensional. Then, by
Again a smooth vector field defined on the Lie bracket of and. The definition means: For any smooth function on
In local coordinates, the vector fields have a representation
Sometimes the Lie bracket is defined with the opposite sign, so instead.
Lie derivative of tensor fields
For a tensor field and a vector field with a local river, the Lie derivative is defined by regarding as
The Lie derivative is linear - in and for a derivation of the tensor algebra firm that is compatible with the contraction. The Lie derivative is characterized and already clearly characterized by their values on functions and vector fields.
In contrast to a context, it is non- linearly.
Properties and Lie algebra
The vector space of all smooth functions with respect to the pointwise multiplication of an algebra. The Lie derivative with respect to a vector field is then a linear derivation, ie it has the properties
- Is linear -
- (Leibniz rule)
Denote the set of all smooth vector fields on, then the Lie derivative is also a linear derivation, and we have:
- (Leibniz rule)
- ( Jacobi identity)
This becomes a Lie algebra.
Definition of the Lie derivative on differential forms
Be a manifold, a vector field, and a differential form. Through evaluation can be a kind of inner product between X and define:
And gets the picture:
This picture has the following characteristics:
- Is R- linearly
- Holds for arbitrary
- Be an arbitrary differential form on M and
Further up the Lie derivative with respect to a vector field for functions defined:
For real differential forms, the Lie derivative is defined with respect to a vector field as follows:
It has the following properties: