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RE: On EulerEquations

• To: mathgroup at smc.vnet.net
• Subject: [mg42210] RE: [mg42180] On EulerEquations
• From: "Wolf, Hartmut" <Hartmut.Wolf at t-systems.com>
• Date: Mon, 23 Jun 2003 05:49:47 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

>-----Original Message-----
>From: alvaro diaz [mailto:alvarodiazfalconi at yahoo.com]
To: mathgroup at smc.vnet.net
>Sent: Sunday, June 22, 2003 2:57 AM
>To: mathgroup at smc.vnet.net
>Subject: [mg42210] [mg42180] On EulerEquations
>
>
><< "Calculus`VariationalMethods`"; 
>
>---- Take a Lagrangian
>
>Subscript[L, i_] = 
>  Subscript[f, i][Subscript[q, i][t]]* D[Subscript[q,
>i][t], t]^2 - Subscript[V, i][Subscript[q, i][t]]
>
>---- Found motions equations
>
>Subscript[eq, i_] = 
>  D[D[Subscript[L, i], D[Subscript[q, i][t],t]], t] -
>D[Subscript[L, i], Subscript[q, i][t]] == 0
>
>---- and ...
>
>EulerEquations[Subscript[L, i],Subscript[q, i][t], t]
>
>---- ok. Now
>
>\[GothicCapitalL] = Sum[Subscript[L, i], {i, 1, n}]
>
>---- then the lagrangian equation is ok
>
>eq = D[D[\[GothicCapitalL], D[Subscript[q, i][t], t]],
>t] - D[\[GothicCapitalL], Subscript[q, i][t]] == 0
>
>---- but
>
>ee = EulerEquations[\[GothicCapitalL], Subscript[q,
>i][t],t]
>
>--- have a bug? If you take
>
>n = 2
>
>---- then EulerEquations works.
>

does it? I get

In[11]:= ee
Out[11]=
Derivative[1][Subscript[f, 1]][Subscript[q, 1][t]]*
    Derivative[1][Subscript[q, 1]][t]^2 + 
   Derivative[1][Subscript[f, 2]][Subscript[q, 2][t]]*
    Derivative[1][Subscript[q, 2]][t]^2 - 
   Derivative[1][Subscript[V, 1]][Subscript[q, 1][t]] - 
   Derivative[1][Subscript[V, 2]][Subscript[q, 2][
     t]] == 0


This is not consistent with your (from above)

>  D[D[Subscript[L, i], D[Subscript[q, i][t],t]], t] -
>D[Subscript[L, i], Subscript[q, i][t]] == 0



Regard:

In[15]:= Attributes[Sum]
Out[15]= {HoldAll, Protected, ReadProtected}

Now define a better Langrangian:

In[17]:=
Subscript[\[GothicCapitalL], s] = Evaluate /@ \[GothicCapitalL]
Out[17]=
Sum[-Subscript[V, i][Subscript[q, i][t]] + 
   Subscript[f, i][Subscript[q, i][t]]*
    Derivative[1][Subscript[q, i]][t]^2, {i, 1, n}]


then also the Euler equations will come out (right):

In[18]:=
EulerEquations[Subscript[\[GothicCapitalL], s], Subscript[q, i][t], t]
Out[18]=
-Sum[D[D[-Subscript[V, i][Subscript[q, i][t]] + 
        Subscript[f, i][Subscript[q, i][t]]*
         Derivative[1][Subscript[q, i]][t]^2, 
       Derivative[1][Subscript[q, i]][t]], t], 
     {i, 1, n}] + 
   Sum[D[-Subscript[V, i][Subscript[q, i][t]] + 
      Subscript[f, i][Subscript[q, i][t]]*
       Derivative[1][Subscript[q, i]][t]^2, 
     Subscript[q, i][t]], {i, 1, n}] == 0

In[19]:= n = 2; %
Out[19]=
(-Derivative[1][Subscript[f, 1]][Subscript[q, 1][t]])*
    Derivative[1][Subscript[q, 1]][t]^2 - 
   Derivative[1][Subscript[f, 2]][Subscript[q, 2][t]]*
    Derivative[1][Subscript[q, 2]][t]^2 - 
   Derivative[1][Subscript[V, 1]][Subscript[q, 1][t]] - 
   Derivative[1][Subscript[V, 2]][Subscript[q, 2][t]] - 
   2*Subscript[f, 1][Subscript[q, 1][t]]*
    Derivative[2][Subscript[q, 1]][t] - 
   2*Subscript[f, 2][Subscript[q, 2][t]]*
    Derivative[2][Subscript[q, 2]][t] == 0


--
Hartmut Wolf