Use of variational operator, 𝛿 in Mathematica

Question

I am working in Hamilton's principle. Part of deriving the equation of motion is to use the delta operator (𝛿) which can be operated just like a differential operator. It is not the function VariationalD in package VariationalMethods.

A post was posted regarding the same subject but it seems that who asked the question found the answer somewhere else. The previous question is here.

Example on the equation is:

T= 0.5 Mu'^2

applying the delta operator

𝛿T=M u' 𝛿 u'

I need to find the first and second variation of this function w.r.t the dependent variable.

regards

Answer 1

Indeed, VariationalD does perform integration by parts which might be unwanted.

For this simple univariat case, one can hack together a quick function myVariationalD:

Clear[δ, myVariationalD]
myVariationalD[expr_, f_[x_]] := Module[{maxDerivative},
  maxDerivative = Max @ Cases[expr, Derivative[m_][f][x] :> m, Infinity];
  D[expr, f[x]] δ[f[x]] + 
   Sum[D[expr, Derivative[m][f][x]] δ[Derivative[m][f][x]], {m,
      1, maxDerivative}]
  ]

The example you have given can then be computed as

expr = 1/2 \[DoubleStruckCapitalM] u'[t]^2;
myVariationalD[expr, u[t]]
(* \[DoubleStruckCapitalM] δ[u'[t]] u'[t] *)