In [1]:

    

using Flows


    

    




INFO: Recompiling stale cache file /home/hofi/.julia/lib/v0.4/Flows.ji for module Flows.

In [2]:

    

@funs F,G,H


    

    
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(F,G,H)

In [3]:

    

@x_vars u,v,w


    

    
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(u,v,w)

In [4]:

    

@t_vars t,s,r


    

    
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(t,s,r)

In [5]:

    

ex = exp(t,D(F),L)*exp(s,G,R)*exp(r,H)


    

    
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$\underbrace{\mathrm{e}^{tD_{F}}}_\mathrm{L}\underbrace{\mathrm{e}^{sD_{G}}}_\mathrm{R}\mathrm{e}^{rD_{H}}$

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show_exponential_labels(false)


    

    
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false

In [7]:

    

ex


    

    
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$\mathrm{e}^{tD_{F}}\mathrm{e}^{sD_{G}}\mathrm{e}^{rD_{H}}$

In [8]:

    

string(ex)


    

    
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"exp(t,D(F),L)*exp(s,D(G),R)*exp(r,D(H))"

In [9]:

    

show_exponential_labels()


    

    
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true

In [10]:

    

ex


    

    
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$\underbrace{\mathrm{e}^{tD_{F}}}_\mathrm{L}\underbrace{\mathrm{e}^{sD_{G}}}_\mathrm{R}\mathrm{e}^{rD_{H}}$

In [11]:

    

ex1=exp(t,D(F),L)


    

    
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$\underbrace{\mathrm{e}^{tD_{F}}}_\mathrm{L}$

In [12]:

    

ex2=exp(t,D(F),R)


    

    
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$\underbrace{\mathrm{e}^{tD_{F}}}_\mathrm{R}$

In [13]:

    

ex1==ex2


    

    
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false

In [14]:

    

ex = ex1+ex2


    

    
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$\underbrace{\mathrm{e}^{tD_{F}}}_\mathrm{L}+\underbrace{\mathrm{e}^{tD_{F}}}_\mathrm{R}$

In [15]:

    

_L = Flows.L
_R = Flows.R


    

    
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Flows.Label("R")

In [16]:

    

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


    

    
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$\underbrace{\mathrm{e}^{tD_{F}}}_\mathrm{L}D_{F}+D_{F}\underbrace{\mathrm{e}^{tD_{F}}}_\mathrm{R}$

In [17]:

    

ex = exp(t,F)*(exp(t,G)+exp(t,H))


    

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

In [18]:

    

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


    

    
Out[18]:





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

In [19]:

    

t_derivative(ex, t)


    

    
Out[19]:





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

In [20]:

    

expand(t_derivative(ex, t))


    

    
Out[20]:





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

In [21]:

    

ex = (exp(t,G)+exp(t,H))*exp(t,F)


    

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

In [22]:

    

t_derivative(ex, t)


    

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

In [23]:

    

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


    

    
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$D_{F}\mathrm{e}^{tD_{G}}D_{F}\mathrm{e}^{tD_{G}}D_{F}\mathrm{e}^{tD_{G}}D_{F}$

In [24]:

    

ex1 = t_derivative(ex, t)


    

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

In [25]:

    

ex1 = D(F)*D(F)*D(G)


    

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

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ex2 = D(F)*D(G)*D(G)


    

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

In [27]:

    

add_factorized(ex1, ex2)


    

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

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add_factorized(D(F), ex2)


    

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

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add_factorized(D(G), ex2)


    

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

In [30]:

    

@funs R


    

    




WARNING: imported binding for R overwritten in module Main






    
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(R,)

In [31]:

    

@t_vars τ


    

    
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(τ,)

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show_exponential_labels()


    

    
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true

In [33]:

    

ex = exp(τ,D(F+R),_L)exp(t-τ,D(F),_R)


    

    
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$\underbrace{\mathrm{e}^{τD_{F+R}}}_\mathrm{L}\underbrace{\mathrm{e}^{(-τ+t)D_{F}}}_\mathrm{R}$

In [34]:

    

show_exponential_labels(false)


    

    
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false

In [35]:

    

ex = t_derivative(ex,τ)


    

    
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$\mathrm{e}^{τD_{F+R}}(D_{F+R}-D_{F})\mathrm{e}^{(-τ+t)D_{F}}$

In [36]:

    

ex = expand_lie_derivatives(ex)


    

    
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$\mathrm{e}^{τD_{F+R}}D_{R}\mathrm{e}^{(-τ+t)D_{F}}$

In [37]:

    

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


    

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

In [38]:

    

t_derivative(ex,t)


    

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

In [39]:

    

add_factorized(2D(F)D(H), -D(F)D(G)D(H))


    

    
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$D_{F}(2\mathrm{Id}-D_{G})D_{H}$

In [40]:

    

ex=2*D(F)*3*D(G)*5*D(H)


    

    
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$30D_{F}D_{G}D_{H}$

In [41]:

    

ex = exp(t,F)D(G)exp(-t,F,_R)


    

    
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$\mathrm{e}^{tD_{F}}D_{G}\mathrm{e}^{-tD_{F}}$

In [42]:

    

ex = t_derivative(ex,t)


    

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

In [43]:

    

ex = t_derivative(ex,t)


    

    
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$\mathrm{e}^{tD_{F}}(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F}))\mathrm{e}^{-tD_{F}}$

In [44]:

    

expand(ex)


    

    
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$\mathrm{e}^{tD_{F}}D_{F}D_{F}D_{G}\mathrm{e}^{-tD_{F}}-2\mathrm{e}^{tD_{F}}D_{F}D_{G}D_{F}\mathrm{e}^{-tD_{F}}+\mathrm{e}^{tD_{F}}D_{G}D_{F}D_{F}\mathrm{e}^{-tD_{F}}$

In [45]:

    

ex = t_derivative(ex,t)


    

    
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$\mathrm{e}^{tD_{F}}(-(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F}))D_{F}+D_{F}(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F})))\mathrm{e}^{-tD_{F}}$

In [46]:

    

expand(ex)


    

    
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$-\mathrm{e}^{tD_{F}}D_{G}D_{F}D_{F}D_{F}\mathrm{e}^{-tD_{F}}-3\mathrm{e}^{tD_{F}}D_{F}D_{F}D_{G}D_{F}\mathrm{e}^{-tD_{F}}+3\mathrm{e}^{tD_{F}}D_{F}D_{G}D_{F}D_{F}\mathrm{e}^{-tD_{F}}+\mathrm{e}^{tD_{F}}D_{F}D_{F}D_{F}D_{G}\mathrm{e}^{-tD_{F}}$

In [47]:

    

ex1=expand_lie_commutators(commutator(D(F),commutator(D(F),commutator(D(F),D(G)))))


    

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

In [48]:

    

ex1 = exp(t,F)*ex1*exp(-t,F,_R)


    

    
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$\mathrm{e}^{tD_{F}}(-(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F}))D_{F}+D_{F}(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F})))\mathrm{e}^{-tD_{F}}$

In [49]:

    

ex==ex1


    

    
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true

In [50]:

    

ex = D(G)*D(F)*exp(2t,F)*D(F)^2*exp(3s,5F)*D(7F)*D(G)


    

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

In [51]:

    

normalize_lie_products(ex)


    

    
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$7D_{G}\mathrm{e}^{(15s+2t)D_{F}}D_{F}D_{F}D_{F}D_{F}D_{G}$

In [52]:

    

normalize_lie_products(ex, to_the_right=[default])


    

    
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$7D_{G}D_{F}D_{F}D_{F}D_{F}\mathrm{e}^{(15s+2t)D_{F}}D_{G}$

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