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