In [1]:

    

using Flows


    

In [2]:

    

@funs F,G,H


    

    
Out[2]:





(F,G,H)

In [3]:

    

@x_vars u,v


    

    
Out[3]:





(u,v)

In [4]:

    

@t_vars t,s


    

    
Out[4]:





(t,s)

In [5]:

    

ex=F+2(F+3G)


    

    
Out[5]:





$6G+3F$

In [6]:

    

exp(2t+3s,D(2F+3G))


    

    
Out[6]:





$\mathrm{e}^{(3s+2t)D_{3G+2F}}$

In [7]:

    

ex = D(3F+2G)


    

    
Out[7]:





$D_{2G+3F}$

In [8]:

    

expand_lie_derivatives(ex)


    

    
Out[8]:





$3D_{F}+2D_{G}$

In [9]:

    

D(-1F)*D(2F)


    

    
Out[9]:





$D_{-F}D_{2F}$

In [10]:

    

exp(2t+3s,D(2F+3G))D(3F+2G)


    

    
Out[10]:





$\mathrm{e}^{(3s+2t)D_{3G+2F}}D_{2G+3F}$

In [11]:

    

ex=D(F)*D(G)


    

    
Out[11]:





$D_{F}D_{G}$

In [12]:

    

typeof(ex)


    

    
Out[12]:





Flows.LieProduct

In [13]:

    

D(H)*ex


    

    
Out[13]:





$D_{H}D_{F}D_{G}$

In [14]:

    

ex=D(F)*(D(G)+D(H))*exp(2t-3s,D(3F))


    

    
Out[14]:





$D_{F}(D_{G}+D_{H})\mathrm{e}^{(-3s+2t)D_{3F}}$

In [15]:

    

string(ex)


    

    
Out[15]:





"D(F)*(D(G)+D(H))*exp(-3s+2t,D(3F))"

In [16]:

    

6D(F)D(H)+6D(F)D(G)


    

    
Out[16]:





$6D_{F}D_{G}+6D_{F}D_{H}$

In [17]:

    

typeof(ex)


    

    
Out[17]:





Flows.LieProduct

In [18]:

    

D(F)D(G)-D(G)D(F)


    

    
Out[18]:





$D_{F}D_{G}-D_{G}D_{F}$

In [19]:

    

typeof(D(F))


    

    
Out[19]:





Flows.LieDerivative

In [20]:

    

exp(-t+17s,D(3F-4G))


    

    
Out[20]:





$\mathrm{e}^{(17s-t)D_{-4G+3F}}$

In [21]:

    

typeof(exp(t,D(F)))


    

    
Out[21]:





Flows.LieExponential

In [22]:

    

-4F+3F


    

    
Out[22]:





$-F$

In [23]:

    

ex


    

    
Out[23]:





$D_{F}(D_{G}+D_{H})\mathrm{e}^{(-3s+2t)D_{3F}}$

In [24]:

    

comb=apply(ex,F(u),u)


    

    
Out[24]:





$D_{F}(D_{G}+D_{H})\mathrm{e}^{(-3s+2t)D_{3F}}F(u)$

In [25]:

    

evaluate(comb)


    

    
Out[25]:





$F''(\mathcal{E}_{3F}(-3s+2t,u))(\partial_{2}\mathcal{E}_{3F}(-3s+2t,u)\cdot G(u),\partial_{2}\mathcal{E}_{3F}(-3s+2t,u)\cdot F(u))+F'(\mathcal{E}_{3F}(-3s+2t,u))\cdot (\partial_{2}^{2}\mathcal{E}_{3F}(-3s+2t,u)(G(u),F(u))+\partial_{2}\mathcal{E}_{3F}(-3s+2t,u)\cdot G'(u)\cdot F(u))+F''(\mathcal{E}_{3F}(-3s+2t,u))(\partial_{2}\mathcal{E}_{3F}(-3s+2t,u)\cdot H(u),\partial_{2}\mathcal{E}_{3F}(-3s+2t,u)\cdot F(u))+F'(\mathcal{E}_{3F}(-3s+2t,u))\cdot (\partial_{2}^{2}\mathcal{E}_{3F}(-3s+2t,u)(H(u),F(u))+\partial_{2}\mathcal{E}_{3F}(-3s+2t,u)\cdot H'(u)\cdot F(u))$

In [26]:

    

-D(F)*G(u)


    

    
Out[26]:





$(-D_{F})G(u)$

In [27]:

    

string(3exp(t,D(F))*u+v)


    

    
Out[27]:





"apply(3exp(t,D(F)),u,u)+v"

In [28]:

    

ex = apply(D(F)*D(G),F(u)+G(u),u)


    

    
Out[28]:





$D_{F}D_{G}[G(\cdot)+F(\cdot)](u)$

In [29]:

    

evaluate(ex)


    

    
Out[29]:





$G''(u)(G(u),F(u))+F'(u)\cdot G'(u)\cdot F(u)+F''(u)(G(u),F(u))+G'(u)\cdot G'(u)\cdot F(u)$

In [30]:

    

comb=apply(D(F), u+2v,u)


    

    
Out[30]:





$D_{F}[(\cdot)+2v](u)$

In [31]:

    

evaluate(comb)


    

    
Out[31]:





$F(u)$

In [32]:

    

string(comb)


    

    
Out[32]:





"apply(D(F),u+2v,u)"

In [33]:

    

ex=D(F+G)


    

    
Out[33]:





$D_{G+F}$

In [34]:

    

typeof(ex.F)


    

    
Out[34]:





Flows.VectorFieldLinearCombination

In [35]:

    

evaluate(D(F),(G(u)+3H(u)),u)


    

    
Out[35]:





$3H'(u)\cdot F(u)+G'(u)\cdot F(u)$

In [36]:

    

evaluate(D(F),(G(H(u))),u)


    

    
Out[36]:





$G'(H(u))\cdot H'(u)\cdot F(u)$

In [37]:

    

evaluate(D(F),u,u)


    

    
Out[37]:





$F(u)$

In [38]:

    

evaluate(D(F+2G),H(u),u)


    

    
Out[38]:





$H'(u)\cdot (2G+F)(u)$

In [39]:

    

ex = D(F)*G(u)


    

    
Out[39]:





$D_{F}G(u)$

In [40]:

    

evaluate(ex)


    

    
Out[40]:





$G'(u)\cdot F(u)$

In [41]:

    

evaluate(exp(t,D(F)), G(u)+3H(u), u)


    

    
Out[41]:





$3H(\mathcal{E}_{F}(t,u))+G(\mathcal{E}_{F}(t,u))$

In [42]:

    

evaluate(exp(t,D(F)), G(H(u)), u)


    

    
Out[42]:





$G(H(\mathcal{E}_{F}(t,u)))$

In [43]:

    

evaluate(exp(t,D(F)), u, u)


    

    
Out[43]:





$\mathcal{E}_{F}(t,u)$

In [44]:

    

evaluate(exp(t+2s,D(3F)), u, u)


    

    
Out[44]:





$\mathcal{E}_{3F}(2s+t,u)$

In [45]:

    

evaluate(exp(t,D(F+G)), u, u)


    

    
Out[45]:





$\mathcal{E}_{G+F}(t,u)$

In [46]:

    

evaluate(D(F)-2D(G),H(u))


    

    
Out[46]:





$H'(u)\cdot F(u)-2H'(u)\cdot G(u)$

In [47]:

    

F(u)-G(v)==F(u)-G(v)


    

    
Out[47]:





true

In [48]:

    

ex = (D(F)*D(G)-D(G)*D(F))*u


    

    
Out[48]:





$(D_{F}D_{G}-D_{G}D_{F})\mathrm{Id}(u)$

In [49]:

    

evaluate(ex)


    

    
Out[49]:





$G'(u)\cdot F(u)-F'(u)\cdot G(u)$

In [50]:

    

ex = exp(t,D(F))*exp(s,D(G))*H(u)


    

    
Out[50]:





$\mathrm{e}^{tD_{F}}\mathrm{e}^{sD_{G}}H(u)$

In [51]:

    

evaluate(ex)


    

    
Out[51]:





$H(\mathcal{E}_{G}(s,\mathcal{E}_{F}(t,u)))$

In [52]:

    

string(ex.lie_ex)


    

    
Out[52]:





"exp(t,D(F))*exp(s,D(G))"

In [53]:

    

G(evaluate_lie_expressions(D(F)*v+v))


    

    
Out[53]:





$G(F(v)+v)$

In [54]:

    

Flows.LieExpressionToSpaceExpressionApplication(exp(t,D(F))*exp(s,D(G)),H(u),u)


    

    
Out[54]:





$\mathrm{e}^{tD_{F}}\mathrm{e}^{sD_{G}}H(u)$

In [55]:

    

evaluate(D(F)*D(F)*exp(t,D(F))*u)


    

    
Out[55]:





$\partial_{2}\mathcal{E}_{F}(t,u)\cdot F'(u)\cdot F(u)+\partial_{2}^{2}\mathcal{E}_{F}(t,u)(F(u),F(u))$

In [56]:

    

evaluate(D(F)*exp(t,D(F))*D(F)*u)


    

    
Out[56]:





$F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot F(u)$

In [57]:

    

evaluate(D(F)*exp(t,D(F))*D(F)*D(F)*u)


    

    
Out[57]:





$F'(\mathcal{E}_{F}(t,u))\cdot F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot F(u)+F''(\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot F(u))$

In [58]:

    

ex =(D(F)+D(G))*(D(F)+D(G))


    

    
Out[58]:





$(D_{F}+D_{G})(D_{F}+D_{G})$

In [59]:

    

expand(ex)


    

    
Out[59]:





$D_{F}D_{F}+D_{F}D_{G}+D_{G}D_{F}+D_{G}D_{G}$

In [60]:

    

ex=(D(F)+D(G))^3


    

    
Out[60]:





$(D_{F}+D_{G})(D_{F}+D_{G})(D_{F}+D_{G})$

In [61]:

    

expand(ex)


    

    
Out[61]:





$D_{G}D_{G}D_{G}+D_{G}D_{F}D_{G}+D_{G}D_{F}D_{F}+D_{F}D_{F}D_{G}+D_{F}D_{F}D_{F}+D_{F}D_{G}D_{F}+D_{G}D_{G}D_{F}+D_{F}D_{G}D_{G}$

In [62]:

    

ex = D(G)^2*exp(t,D(F))*D(F)^2*exp(t,D(G))*u


    

    
Out[62]:





$D_{G}D_{G}\mathrm{e}^{tD_{F}}D_{F}D_{F}\mathrm{e}^{tD_{G}}\mathrm{Id}(u)$

In [63]:

    

evaluate(ex)


    

    
Out[63]:





$4\partial_{2}^{3}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+\partial_{2}^{3}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),F(\mathcal{E}_{F}(t,u)),\partial_{2}^{2}\mathcal{E}_{F}(t,u)(G(u),G(u))+\partial_{2}\mathcal{E}_{F}(t,u)\cdot G'(u)\cdot G(u))+\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot F(\mathcal{E}_{F}(t,u)),\partial_{2}^{2}\mathcal{E}_{F}(t,u)(G(u),G(u))+\partial_{2}\mathcal{E}_{F}(t,u)\cdot G'(u)\cdot G(u))+2\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F''(\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+F'(\mathcal{E}_{F}(t,u))\cdot F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+\partial_{2}^{3}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+2\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F''(\mathcal{E}_{F}(t,u))(\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+F'(\mathcal{E}_{F}(t,u))\cdot (\partial_{2}^{2}\mathcal{E}_{F}(t,u)(G(u),G(u))+\partial_{2}\mathcal{E}_{F}(t,u)\cdot G'(u)\cdot G(u)),F(\mathcal{E}_{F}(t,u)))+\partial_{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))\cdot (F'''(\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+F''(\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),\partial_{2}^{2}\mathcal{E}_{F}(t,u)(G(u),G(u))+\partial_{2}\mathcal{E}_{F}(t,u)\cdot G'(u)\cdot G(u))+2F''(\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+F'(\mathcal{E}_{F}(t,u))\cdot (F''(\mathcal{E}_{F}(t,u))(\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+F'(\mathcal{E}_{F}(t,u))\cdot (\partial_{2}^{2}\mathcal{E}_{F}(t,u)(G(u),G(u))+\partial_{2}\mathcal{E}_{F}(t,u)\cdot G'(u)\cdot G(u))))+2\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+\partial_{2}^{4}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))$

In [64]:

    

expand(evaluate(ex))


    

    
Out[64]:





$\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot F(\mathcal{E}_{F}(t,u)),\partial_{2}^{2}\mathcal{E}_{F}(t,u)(G(u),G(u)))+2\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F''(\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+\partial_{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))\cdot F'(\mathcal{E}_{F}(t,u))\cdot F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}^{2}\mathcal{E}_{F}(t,u)(G(u),G(u))+2\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F''(\mathcal{E}_{F}(t,u))(\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u)),F(\mathcal{E}_{F}(t,u)))+\partial_{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))\cdot F''(\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G'(u)\cdot G(u))+\partial_{2}^{4}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+2\partial_{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))\cdot F''(\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G'(u)\cdot G(u))+2\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+\partial_{2}^{3}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G'(u)\cdot G(u))+\partial_{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))\cdot F'(\mathcal{E}_{F}(t,u))\cdot F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G'(u)\cdot G(u)+\partial_{2}^{3}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),F(\mathcal{E}_{F}(t,u)),\partial_{2}^{2}\mathcal{E}_{F}(t,u)(G(u),G(u)))+\partial_{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))\cdot F''(\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),\partial_{2}^{2}\mathcal{E}_{F}(t,u)(G(u),G(u)))+2\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+\partial_{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))\cdot F'(\mathcal{E}_{F}(t,u))\cdot F''(\mathcal{E}_{F}(t,u))(\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+4\partial_{2}^{3}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+2\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot G'(u)\cdot G(u),F(\mathcal{E}_{F}(t,u)))+2\partial_{2}^{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}^{2}\mathcal{E}_{F}(t,u)(G(u),G(u)),F(\mathcal{E}_{F}(t,u)))+\partial_{2}^{3}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))(F'(\mathcal{E}_{F}(t,u))\cdot F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))+\partial_{2}\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))\cdot F'''(\mathcal{E}_{F}(t,u))(F(\mathcal{E}_{F}(t,u)),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u),\partial_{2}\mathcal{E}_{F}(t,u)\cdot G(u))$

In [65]:

    

C = commutator


    

    
Out[65]:





commutator (generic function with 4 methods)

In [66]:

    

ex = C(5D(F)+7D(H),2D(G)+3D(H))


    

    
Out[66]:





$[5D_{F}+7D_{H},2D_{G}+3D_{H}]$

In [67]:

    

expand(ex)


    

    
Out[67]:





$14[D_{H},D_{G}]+15[D_{F},D_{H}]+10[D_{F},D_{G}]$

In [68]:

    

ex = (11exp(t,F)+13exp(t,G))*C(5D(F)+7D(H),2D(G)+3D(H))


    

    
Out[68]:





$(11\mathrm{e}^{tD_{F}}+13\mathrm{e}^{tD_{G}})[5D_{F}+7D_{H},2D_{G}+3D_{H}]$

In [69]:

    

expand(ex, expand_commutators=false)


    

    
Out[69]:





$11\mathrm{e}^{tD_{F}}[5D_{F}+7D_{H},2D_{G}+3D_{H}]+13\mathrm{e}^{tD_{G}}[5D_{F}+7D_{H},2D_{G}+3D_{H}]$

In [70]:

    

ex = expand(ex)


    

    
Out[70]:





$195\mathrm{e}^{tD_{G}}[D_{F},D_{H}]+110\mathrm{e}^{tD_{F}}[D_{F},D_{G}]+154\mathrm{e}^{tD_{F}}[D_{H},D_{G}]+165\mathrm{e}^{tD_{F}}[D_{F},D_{H}]+182\mathrm{e}^{tD_{G}}[D_{H},D_{G}]+130\mathrm{e}^{tD_{G}}[D_{F},D_{G}]$

In [71]:

    

ex = merge_lie_derivatives(ex)


    

    
Out[71]:





$165\mathrm{e}^{tD_{F}}D_{[H,F]}+D_{0}+195\mathrm{e}^{tD_{G}}D_{[H,F]}+182\mathrm{e}^{tD_{G}}D_{[G,H]}+130\mathrm{e}^{tD_{G}}D_{[G,F]}+110\mathrm{e}^{tD_{F}}D_{[G,F]}+154\mathrm{e}^{tD_{F}}D_{[G,H]}$

In [72]:

    

expand(ex, expand_commutators=false)


    

    
Out[72]:





$165\mathrm{e}^{tD_{F}}D_{[H,F]}+D_{0}+195\mathrm{e}^{tD_{G}}D_{[H,F]}+182\mathrm{e}^{tD_{G}}D_{[G,H]}+130\mathrm{e}^{tD_{G}}D_{[G,F]}+110\mathrm{e}^{tD_{F}}D_{[G,F]}+154\mathrm{e}^{tD_{F}}D_{[G,H]}$

In [73]:

    

ex = (C(D(F),C(D(G),D(H))) + C(D(G),C(D(H),D(F))) + C(D(H),C(D(F),D(G))))*u


    

    
Out[73]:





$([D_{G},[D_{H},D_{F}]]+[D_{F},[D_{G},D_{H}]]+[D_{H},[D_{F},D_{G}]])\mathrm{Id}(u)$

In [74]:

    

merge_lie_derivatives(ex)


    

    
Out[74]:





$D_{[[G,F],H]+[[F,H],G]+[[H,G],F]}\mathrm{Id}(u)$

In [75]:

    

merge_lie_derivatives(ex, merge_linear_combinations=false)


    

    
Out[75]:





$(D_{[[H,G],F]}+D_{[[G,F],H]}+D_{[[F,H],G]})\mathrm{Id}(u)$

In [76]:

    

ex = expand_lie_commutators(ex)


    

    
Out[76]:





$(-(D_{G}D_{H}-D_{H}D_{G})D_{F}-(D_{H}D_{F}-D_{F}D_{H})D_{G}+D_{H}(D_{F}D_{G}-D_{G}D_{F})+D_{F}(D_{G}D_{H}-D_{H}D_{G})-(D_{F}D_{G}-D_{G}D_{F})D_{H}+D_{G}(D_{H}D_{F}-D_{F}D_{H}))\mathrm{Id}(u)$

In [77]:

    

ex = expand_lie_expressions(ex)


    

    
Out[77]:





$(0)\mathrm{Id}(u)$

In [78]:

    

evaluate_lie_expressions(ex)


    

    
Out[78]:





$0$

In [79]:

    

ex=F(u,D(F)*u)


    

    
Out[79]:





$F'(u)\cdot D_{F}\mathrm{Id}(u)$

In [80]:

    

evaluate_lie_expressions(ex)


    

    
Out[80]:





$F'(u)\cdot F(u)$

In [81]:

    

evaluate_lie_expressions(ex)


    

    
Out[81]:





$F'(u)\cdot F(u)$

In [82]:

    

ex = Flows.commutator(D(F),D(F)+D(G))


    

    
Out[82]:





$[D_{F},D_{F}+D_{G}]$

In [83]:

    

LieProduct(LieExpression[])


    

    
Out[83]:





$\mathrm{Id}$

In [84]:

    

lie_id


    

    
Out[84]:





$\mathrm{Id}$

In [85]:

    

ex = exp(t,D(C(F,G)))*u


    

    
Out[85]:





$\mathrm{e}^{tD_{[F,G]}}\mathrm{Id}(u)$

In [86]:

    

evaluate(ex)


    

    
Out[86]:





$\mathcal{E}_{[F,G]}(t,u)$

In [87]:

    

ex = D(commutator(F,commutator(F,commutator(F,commutator(F,G)))))*u


    

    
Out[87]:





$D_{[F,[F,[F,[F,G]]]]}\mathrm{Id}(u)$

In [88]:

    

ex = expand_lie_derivatives(ex)


    

    
Out[88]:





$[[[[D_{G},D_{F}],D_{F}],D_{F}],D_{F}]\mathrm{Id}(u)$

In [89]:

    

merge_lie_derivatives(ex)


    

    
Out[89]:





$D_{[F,[F,[F,[F,G]]]]}\mathrm{Id}(u)$

In [90]:

    

ex = D(F+G)*D(C(F,G))


    

    
Out[90]:





$D_{G+F}D_{[F,G]}$

In [91]:

    

ex = expand_lie_derivatives(ex, expand_commutators=false)


    

    
Out[91]:





$(D_{F}+D_{G})D_{[F,G]}$

In [92]:

    

ex = expand_lie_derivatives(ex, expand_linear_combinations=false)


    

    
Out[92]:





$(D_{F}+D_{G})[D_{G},D_{F}]$

In [93]:

    

ex = expand_lie_commutators(ex)


    

    
Out[93]:





$(D_{F}+D_{G})(-D_{F}D_{G}+D_{G}D_{F})$

In [94]:

    

expand(ex)


    

    
Out[94]:





$-D_{G}D_{F}D_{G}-D_{F}D_{F}D_{G}+D_{F}D_{G}D_{F}+D_{G}D_{G}D_{F}$

In [95]:

    

ex=D(3F+5commutator(F,G)+commutator(7F,commutator(3G,11H)))


    

    
Out[95]:





$D_{5[F,G]+[7F,[3G,11H]]+3F}$

In [96]:

    

ex1 = expand_lie_derivatives(ex)


    

    
Out[96]:





$[[11D_{H},3D_{G}],7D_{F}]+3D_{F}+5[D_{G},D_{F}]$

In [97]:

    

merge_lie_derivatives(ex1)


    

    
Out[97]:





$D_{5[F,G]+[7F,[3G,11H]]+3F}$

In [98]:

    

ex1=expand_lie_derivatives(ex, expand_commutators=false)


    

    
Out[98]:





$3D_{F}+D_{[7F,[3G,11H]]}+5D_{[F,G]}$

In [99]:

    

merge_lie_derivatives(ex1)


    

    
Out[99]:





$D_{5[F,G]+[7F,[3G,11H]]+3F}$

In [100]:

    

expand_lie_derivatives(ex, expand_linear_combinations=false)


    

    
Out[100]:





$D_{5[F,G]+[7F,[3G,11H]]+3F}$

In [101]:

    

@funs G1,G2,G3,G4
@t_vars t1,t2,t3,t4


    

    
Out[101]:





(t1,t2,t3,t4)

In [102]:

    

ex1=exp(t1,D(G1))*exp(t2,D(G2))*D(H)*exp(t3,D(G3))*exp(t4,D(G4))*v


    

    
Out[102]:





$\mathrm{e}^{t1D_{G1}}\mathrm{e}^{t2D_{G2}}D_{H}\mathrm{e}^{t3D_{G3}}\mathrm{e}^{t4D_{G4}}\mathrm{Id}(v)$

In [103]:

    

ex1=evaluate(ex1)


    

    
Out[103]:





$\partial_{2}\mathcal{E}_{G4}(t4,\mathcal{E}_{G3}(t3,\mathcal{E}_{G2}(t2,\mathcal{E}_{G1}(t1,v))))\cdot \partial_{2}\mathcal{E}_{G3}(t3,\mathcal{E}_{G2}(t2,\mathcal{E}_{G1}(t1,v)))\cdot H(\mathcal{E}_{G2}(t2,\mathcal{E}_{G1}(t1,v)))$

In [104]:

    

ex2=E(G4,t4,E(G3,t3,u),E(G3,t3,u,H(u)))


    

    
Out[104]:





$\partial_{2}\mathcal{E}_{G4}(t4,\mathcal{E}_{G3}(t3,u))\cdot \partial_{2}\mathcal{E}_{G3}(t3,u)\cdot H(u)$

In [105]:

    

ex2=substitute(ex2,u,E(G2,t2,E(G1,t1,v)))


    

    
Out[105]:





$\partial_{2}\mathcal{E}_{G4}(t4,\mathcal{E}_{G3}(t3,\mathcal{E}_{G2}(t2,\mathcal{E}_{G1}(t1,v))))\cdot \partial_{2}\mathcal{E}_{G3}(t3,\mathcal{E}_{G2}(t2,\mathcal{E}_{G1}(t1,v)))\cdot H(\mathcal{E}_{G2}(t2,\mathcal{E}_{G1}(t1,v)))$

In [106]:

    

ex1==ex2


    

    
Out[106]:





true

In [107]:

    

ex = exp(t,D(F))*D(F)*u


    

    
Out[107]:





$\mathrm{e}^{tD_{F}}D_{F}\mathrm{Id}(u)$

In [108]:

    

evaluate(ex)


    

    
Out[108]:





$F(\mathcal{E}_{F}(t,u))$

In [109]:

    

ex = D(F)*exp(t,D(F))*u


    

    
Out[109]:





$D_{F}\mathrm{e}^{tD_{F}}\mathrm{Id}(u)$

In [110]:

    

evaluate(ex)


    

    
Out[110]:





$\partial_{2}\mathcal{E}_{F}(t,u)\cdot F(u)$

In [111]:

    

ex = F(D(commutator(F,G))*u)


    

    
Out[111]:





$F(D_{[F,G]}\mathrm{Id}(u))$

In [112]:

    

ex = expand_lie_derivatives(ex)


    

    
Out[112]:





$F([D_{G},D_{F}]\mathrm{Id}(u))$

In [113]:

    

ex = evaluate_lie_expressions(ex)


    

    
Out[113]:





$F(-G'(u)\cdot F(u)+F'(u)\cdot G(u))$

In [114]:

    

ex = exp(2s+t,D(F))


    

    
Out[114]:





$\mathrm{e}^{(2s+t)D_{F}}$

In [115]:

    

t_derivative(ex,t)


    

    
Out[115]:





$\mathrm{e}^{(2s+t)D_{F}}D_{F}$

In [116]:

    

t_derivative(ex,s)


    

    
Out[116]:





$2\mathrm{e}^{(2s+t)D_{F}}D_{F}$

In [117]:

    

t_derivative(ex,s, to_the_right=[default])


    

    
Out[117]:





$2D_{F}\mathrm{e}^{(2s+t)D_{F}}$

In [118]:

    

ex = 5exp(2t,F)+7exp(3t,G)


    

    
Out[118]:





$7\mathrm{e}^{3tD_{G}}+5\mathrm{e}^{2tD_{F}}$

In [119]:

    

t_derivative(ex,t)


    

    
Out[119]:





$10\mathrm{e}^{2tD_{F}}D_{F}+21\mathrm{e}^{3tD_{G}}D_{G}$

In [120]:

    

ex = D(F)*exp(2t,D(G))*D(F)*exp(3t,D(H))*D(F)


    

    
Out[120]:





$D_{F}\mathrm{e}^{2tD_{G}}D_{F}\mathrm{e}^{3tD_{H}}D_{F}$

In [121]:

    

t_derivative(ex,t)


    

    
Out[121]:





$D_{F}(3\mathrm{e}^{2tD_{G}}D_{F}\mathrm{e}^{3tD_{H}}D_{H}+2\mathrm{e}^{2tD_{G}}D_{G}D_{F}\mathrm{e}^{3tD_{H}})D_{F}$

In [122]:

    

t_derivative(ex,t, to_the_right=[default])


    

    
Out[122]:





$D_{F}(2D_{G}\mathrm{e}^{2tD_{G}}D_{F}+3\mathrm{e}^{2tD_{G}}D_{F}D_{H})\mathrm{e}^{3tD_{H}}D_{F}$

In [123]:

    

ex=commutator(exp(t,F),exp(t,G))


    

    
Out[123]:





$[\mathrm{e}^{tD_{F}},\mathrm{e}^{tD_{G}}]$

In [124]:

    

t_derivative(ex, t)


    

    
Out[124]:





$[\mathrm{e}^{tD_{F}},\mathrm{e}^{tD_{G}}D_{G}]+[\mathrm{e}^{tD_{F}}D_{F},\mathrm{e}^{tD_{G}}]$

In [125]:

    

ex=D(G)*commutator(exp(t,F),D(G))*D(G)


    

    
Out[125]:





$D_{G}[\mathrm{e}^{tD_{F}},D_{G}]D_{G}$

In [126]:

    

t_derivative(ex, t)


    

    
Out[126]:





$D_{G}[\mathrm{e}^{tD_{F}}D_{F},D_{G}]D_{G}$

In [ ]: