6.6. Lie derivatives

xTensor` does not define independent Lie derivatives, but a single command LieD which depends both on a contravariant vector field and on the expression that we want to derive.

LieD                    Lie derivative
LieDToCovD                Expansion of Lie derivative in terms of covariant derivatives

Lie derivatives.

Lie derivatives of expressions with respect to a contravariant vector field are not automatically expanded in terms of derivatives.

In[497]:=

LieD[v[a]][T[-b, c]]

Out[497]=

ℒ_vT_b ^( c)

However, we can expand them using any derivative (PD is the default).

In[498]:=

LieDToCovD[LieD[w[a]][T[-b, c]], PD]

Out[498]=

w_ ^a ∂_a^ T_b ^( c) - T_b ^( a) ∂_a^ w_ ^c + T_a ^( c) ∂_b^ w_ ^a

In[499]:=

LieDToCovD[LieD[w[a]][T[-b, c]], Cd]

Out[499]=

w_ ^a (▽_a^ T_b ^( c)) - T_b ^( a) (▽_a^ w_ ^c) + T_a ^( c) (▽_b^ w_ ^a)

The derivative can also be expanded directly:

In[500]:=

LieD[w[a], Cd][T[-b, c]]

Out[500]=

w_ ^a (▽_a^ T_b ^( c)) - T_b ^( a) (▽_a^ w_ ^c) + T_a ^( c) (▽_b^ w_ ^a)

LieD knows the Leibniz rule:

In[501]:=

LieD[w[a]][T[a] T[-a] ]

Out[501]=

T_ ^a (ℒ_wT_a^ ) + T_a^ (ℒ_wT_ ^a)

We can have any expression in the "directional argument". Recall however that it is not a tensorial slot (it hence cannot be described with an abstract index):

In[502]:=

LieD[7r[] w[a] + v[a]][T[a]]

Out[502]=

ℒ_vT_ ^a + 7 (ℒ_ (r_^ w_ ^a) T_ ^a)

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