(* ::Package:: *)

(* ::Input:: *)

(*Needs["VariationalMethods`"];*)

(**)

(*SimpleFunctionCase[x_]:=y[x]^2+2y'[x]^2;*)

(*HighOrderFunctionCase[x_]:=y[x]^2+2y'[x]^2+y''[x]^2;*)

(*MultiFunctionCase[x_]:=y'[x]^2+z'[x]^2+2y[x];*)

(*MultiVariableCase[x_,y_,z_]:=D[u[x,y,z],x]^2+D[u[x,y,z],y]^2+D[u[x,y,z],z]^2+2u[x,y,z] f[x,y,z];*)

(**)

(*(*The variationalD and EulerEquations method can handle all high order variation, multivariable and multifunction*)*)

(*VariationalD[HighOrderFunctionCase[x],y[x],x]*)

(*VariationalD[MultiFunctionCase[x],{y[x],z[x]},x]*)

(*VariationalD[MultiVariableCase[x,y,z],u[x,y,z],{x,y,z}]*)

(**)

(*EulerEquations[HighOrderFunctionCase[x],y[x],x]*)

(*EulerEquations[MultiFunctionCase[x],{y[x],z[x]},x]*)

(*EulerEquations[MultiVariableCase[x,y,z],u[x,y,z],{x,y,z}]*)

(**)

(*(*For splitting the calculation process*)*)

(*SimpleVariation[F_]:=D[F,y[x]]-D[D[F,y'[x]],x];*)

(*HighOrderVariation[F_]:=D[F,y[x]]-D[D[F,y'[x]],x]+D[D[F,y''[x]],{x,2}];*)

(*MultiFunctionVariation[F_]:={D[F,y[x]]-D[D[F,y'[x]],x],D[F,z[x]]-D[D[F,z'[x]],x]};*)

(*MultiVariableVariation[F_]:=D[F,u[x,y,z]]-D[D[F,D[u[x,y,z],x]],x]-D[D[F,D[u[x,y,z],y]],y]-D[D[F,D[u[x,y,z],z]],z];*)

(**)

(*SimpleVariation[SimpleFunctionCase[x]]*)

(*HighOrderVariation[HighOrderFunctionCase[x]]*)

(*MultiFunctionVariation[MultiFunctionCase[x]]*)

(*MultiVariableVariation[MultiVariableCase[x,y,z]]*)

(**)

(*(*Isoperimetric problem F is Variational Function and G is the Isoperimetric Constraint*)*)

(* F[x_]:=y'[x]^2+x^2;*)

(*G[x_]:=y[x]^2;*)

(*EulerEquations[F[x]+lambda G[x],y[x],x]*)

(**)

(*(*Weierstrass Form of Euler Equation*)*)

(*F[t_]:=(y'[t]^2-y[t]^2 x'[t]^2)/x'[t];*)

(*Simplify[D[F[t],x[t],y'[t]]-D[F[t],x'[t],y[t]]+(x'[t] y''[t]-x''[t] y'[t])D[F[t],x'[t],y'[t]]/(y'[t] x'[t])]*)

(**)

(*(*Non Weierstrass Form of Euler Equation*)*)

(*D[F[t],x[t]]-D[D[F[t],x'[t],t]]*)

(*D[F[t],y[t]]-D[D[F[t],y'[t],t]]*)