Lie derivative of a 2-form

$ \omega^2 のLie微分は$ L_{u} \omega^2_{ij} \xi^i \eta^j $ = u^k \partial_k (\omega^2_{ij} \xi^i \eta^j) $ = u^k (\partial_k \omega^2_{ij}) \xi^i \eta^j + \omega^2_{ij} (u^k \partial_k \xi^i) \eta^j + \omega^2_{ij} \xi^i (u^k \partial_k \eta^j)

$ = (u^k \partial_k \omega^2_{ij}) \xi^i \eta^j + \omega^2_{ij} (u^k \partial_k \xi^i - \xi^k \partial_k u^i) + \omega^2_{ij} \xi^k (\partial_k u^i) \eta^j + \omega^2_{ij} \xi^i (u^k \partial_k \eta^j - \eta^k \partial_k u^j) + \omega^2_{ij} \xi^i \eta^k \partial_k u^j

$ = (u^k \partial_k \omega^2_{ij} + \omega^2_{lj} \partial_i u^l + \omega^2_{im} \partial_j u^m) \xi^i \eta^j + \omega^2_{ij} (u^k \partial_k \xi^i - \xi^k \partial_k u^i) + \omega^2_{ij} \xi^i (u^k \partial_k \eta^j - \eta^k \partial_k u^j)

3次元：$ \omega^2 := \epsilon_{ijk} \Omega^i dq^j \wedge dq^k

$ (L_u \omega^2)(\xi,\eta) $ = (u^k \partial_k \epsilon_{ijn} \Omega^n + \epsilon_{ljn} \Omega^n \partial_i u^l + \epsilon_{iln} \Omega^n \partial_j u^l) \xi^i \eta^j

ex) $ = (u^k \partial_k \epsilon_{123} \Omega^3 + \epsilon_{l2n} \Omega^n \partial_1 u^l + \epsilon_{1ln} \Omega^n \partial_2 u^l) \xi^1 \eta^2

$ = (u^k \partial_k \epsilon_{123} \Omega^3 + \epsilon_{123} \Omega^3 \partial_1 u^1 + \epsilon_{321} \Omega^1 \partial_1 u^3 + \epsilon_{123} \Omega^3 \partial_2 u^2 + \epsilon_{132} \Omega^2 \partial_2 u^3) \xi^1 \eta^2

$ = (u^k \partial_k \Omega^3 + \Omega^3 \partial_1 u^1 - \Omega^1 \partial_1 u^3 + \Omega^3 \partial_2 u^2 - \Omega^2 \partial_2 u^3) \xi^1 \eta^2

$ = (u^k \partial_k \Omega^3 + \Omega^3 \partial_1 u^1 + \Omega^3 \partial_2 u^2 + \Omega^3 \partial_3 u^3 - \Omega^1 \partial_1 u^3 - \Omega^2 \partial_2 u^3 - \Omega^3 \partial_3 u^3) \xi^1 \eta^2

$ = (u^k \partial_k \Omega^3 + \Omega^3 \partial_k u^k - \Omega^k \partial_k u^3) \xi^1 \eta^2 $ = (\partial_k (u^k \Omega^3) - \Omega^k \partial_k u^3) \xi^1 \eta^2

多分$ L_u \omega^2 = \epsilon_{ijk} ( \partial_m (u^m \Omega^i) - \Omega^m \partial_m u^i ) dq^j \wedge dq^k

$ F_{jk}(X(a,t),t) \, dX^j(a,t) \wedge dX^k(a,t) $ = F_{jk}(X(a,t),t) \left.\frac{\partial X^j}{\partial q^l}\right|_{q=a} da^l \wedge \left.\frac{\partial X^k}{\partial q^m}\right|_{q=a} da^m

ここで$ K = \frac12 \rho |\bm U|^2 = \frac12 \rho g_{ij} u^i u^j ,$ \omega^2 := \frac{\delta K}{\delta \bm U} ,$ \bm U = g_{ij} u^i dq^j ならば$ \Omega^j = \rho u^j なので$ L_u \omega^2 = \epsilon_{ijk} ( \partial_m (u^m \underbrace{\rho u^i}_{\Omega^i}) - \underbrace{\rho u^m}_{\Omega^m} \partial_m u^i ) dq^j \wedge dq^k $ = \partial_m ( \rho u^m) \epsilon_{ijk} u^i dq^j \wedge dq^k

Lie derivative of a 1-form

$ \omega^1 のLie微分は$ L_{u} \omega^1_i \xi^i $ = u^k \partial_k (\omega^1_i \xi^i) $ = u^k (\partial_k \omega^1_i) \xi^i + \omega^1_i (u^k \partial_k \xi^i)