atlas[L] - Lie derivative 

Calling Sequence: 

    L[V1, V2, V3..., Vn](expr)
    L(V1, V2, V3..., Vn, expr)
    atlas[L](V1, V2, V3..., Vn, expr) 

Parameters: 

       expr - any expression (on which Lie derivative is defined).
       V1, V2, V3..., Vn - vector fields. 

Description: 

• The L - procedure allows one to calculate the Lie derivative on an expression along given vector field. The derivative has the following properties.

 

• For any vector fields and expression we have:

 

• For any vector field X and 0-form we have:   

 

• For vector fields X and Y we have:   

 

• For any vector field X and tensor fields and the Leibniz rule for the Lie derivative takes place:

 

• For any vector field X and p-form we have:

 

Examples: 

>	restart: with(atlas):

 

Declare constants:  

>	Constants(Lambda);

 

(2.1)

 

Declare functions:  

>	Functions(F=F(x,y));

 

(2.2)

 

Declare p-forms:  

>	Forms(omega=p);

 

(2.3)

 

Declare vectors:  

>	Vectors(X,Y,Z);

 

(2.4)

 

Declare tensors:  

>	Tensors(T=[n,k],Omega=[l,m]);

 

(2.5)

 

Using L- procedure: 

Just definition for "long" Lie operator:
'L[X,Y,Z](T)'=L[X,Y,Z](T);
'L(X,Y,Z,T)'=L(X,Y,Z,T);
'L[X](Y,Z,T)'=L[X](Y,Z,T); 

 

 

	(2.6)

 

As h is 0-form by defaults then:
'L[X](h)'=L[X](h); 

(2.7)

 

F- declared as function F=F(x,y) thus:
'L[X]'(F)=L[X](F);
 

(2.8)

 

As declared as constant thus:
'L[X]'(Lambda)=L[X](Lambda); 

(2.9)

 

Lie derivative is linear with respect to any argument:
'L[X]'(Y+Z)=L[X](Y+Z); 

(2.10)

 

And:
'L[X+Y]'(Z)=L[X+Y](Z); 

(2.11)

 

More complex examples:
'L[f*X+h*Y]'(Z)=L[f*X+h*Y](Z); 

(2.12)

 

And:
'L[Z]'(f*X+h*Y)=L[Z](f*X+h*Y); 

(2.13)

 

 - declared as p-form thus:
'L[X]'(omega)=L[X](omega); 

(2.14)

 

Verify that Leibniz rule takes place for tensor product:
'L[X]'(T&.Omega)=L[X](T&.Omega); 

(2.15)

 

>

 

See Also:  

atlas, atlas[d], atlas[cov], atlas[`&.`], atlas[`&^`], atlas[iota].