Function Repository Resource:

AllDependentVariables

Source Notebook

Retrieve a list of all dependent variables for a given expression

Contributed by: E. Chan-López, Jaime Manuel Cabrera & Jorge Mauricio Paulin Fuentes

ResourceFunction["AllDependentVariables"][expr,ivar]

gives all variables dependent on ivar within a given expr.

Details and Options

ResourceFunction["AllDependentVariables"] threads over lists in the first argument.

Similar to Variables, ResourceFunction["AllDependentVariables"] also has the Modulus option.

Examples

Basic Examples (1)

Use AllDependentVariables to get the expression that matches to be a mathematical solution:

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Scope (8)

Use AllDependentVariables to get the expressions that match dependent variable:

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Use AllDependentVariables with a list of complicated expressions:

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$ResourceFunction[ "AllDependentVariables", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{Exp[x[t, s]] + 2 w[t, s], 1/2 m2 (x2'[t, s]^2 + y2'[t, s]^2 + z2'[t, s]^2)}, {t, s}]$

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Use AllDependentVariables with an inhomogeneous first-order ordinary differential equation:

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Use AllDependentVariables with a differential equation with a piecewise coefficient:

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Use AllDependentVariables with a system of delay differential equations:

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$ResourceFunction[ "AllDependentVariables", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{x'[t] == a y[t - 1] + y[t - 3], y'[t] == x[t - 1], x[t /; t <= 0] == t, y[t /; t <= 0] == t^2}, t]$

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Use AllDependentVariables with a Caputo fractional differential equation of order 1/2:

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Use AllDependentVariables with a linear first-order partial differential equation:

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Use AllDependentVariables with a singular Abel integral equation:

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$ResourceFunction[ "AllDependentVariables", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][Sin[a x] == \!$ \*SubsuperscriptBox[\(\[Integral]$, $0$, $x$]$ \*FractionBox[\(y[t]$, SqrtBox[$x - t$]] \[DifferentialD]t\)\), t]$

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Options (2)

Modulus (2)

Find dependent variables present after reducing coefficients modulo 2:

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For polynomials, AllDependentVariables and Variables gives the same results:

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Applications (3)

Use AllDependentVariables to define a simple function to compute the equations of motion for a given Hamiltonian:

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$HamiltonianEqns[H_, time_Symbol] := Module[{sols, classcoordsprev, classcoords, momentaeqns, velseqns}, sols = ResourceFunction[ "AllDependentVariables", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][H, time]; classcoordsprev = GatherBy[sols, FreeQ[Characters[ToString[Head[#]]], "p"] &]; classcoords = Which[Length[classcoordsprev] == 1, Join @@ List[classcoordsprev, Transpose[ Map[ToExpression@*StringJoin, Map[DeleteCases[#, "p"] &@*Characters@*ToString, Transpose@classcoordsprev]]]], Length[classcoordsprev] == 2 && Length[classcoordsprev[[1]]] === Length[classcoordsprev[[2]]], classcoordsprev, Length[classcoordsprev] == 2 && Length[classcoordsprev[[1]]] =!= Length[classcoordsprev[[2]]], Join @@ List[{classcoordsprev[[1]]}, {Transpose[ Map[ToExpression@*StringJoin, Map[DeleteCases[#, "p"] &@*Characters@*ToString, Transpose@classcoordsprev[[1]]]]]}]]; momentaeqns = Thread[D[classcoords[[1]], time] == Simplify[Map[-D[H, #] &, classcoords[[2]]]]]; velseqns = Thread[D[classcoords[[2]], time] == Simplify[Map[D[H, #] &, classcoords[[1]]]]]; Flatten[ Transpose[List[velseqns, momentaeqns]]] /. {Csc[any_]^2 :> 1/HoldForm[Sin][any]^2, Cot[any_] :> HoldForm[Cos][any]/HoldForm[Sin][any]}]$

Use HamiltonianEqns with the Hamiltonian for the spherical pendulum:

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$H1 = -g l m Cos[\[Theta][t]] + p\[Theta][t]^2/(2 l^2 m) + p\[Phi][t]^2/(2 l^2 m Sin[\[Theta][t]]^2);$

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Use HamiltonianEqns with the Hamiltonian for the PUMA-Like Robot:

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$H2 = g (m2 z2 + m3 z3) + px2[t]^2/(2 m2) + px3[t]^2/(2 m3) + py2[t]^2/(2 m2) + py3[t]^2/(2 m3) + pz2[t]^2/(2 m2) + pz3[t]^2/( 2 m3) + p\[Theta]2[t]^2/(2 A2) + p\[Theta]3[t]^2/(2 A3) + p\[Psi]1[ t]^2/(2 (C1 + C2 Cos[\[Theta]2[t]]^2 + C3 Cos[\[Theta]3[t]]^2 + B2 Sin[\[Theta]2[t]]^2 + B3 Sin[\[Theta]3[t]]^2));$

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Properties and Relations (3)

Use AllDependentVariables with the resource function SymbolToSubscript:

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Use AllDependentVariables with the resource functions FormalizeSymbols and SolutionRulesToFunctions:

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$L = 1/2 m l^2 (\[Theta]'[t]^2 + Sin[\[Theta][t]]^2*\[Phi]'[t]^2) + m g l Cos[\[Theta][t]];$

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L /. ResourceFunction["SolutionRulesToFunctions"][
Thread[# -> Through[ReleaseHold@
HoldForm[ResourceFunction]["FormalizeSymbols"][Head /@ #][
t]]]] &@ResourceFunction[
"AllDependentVariables", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][L, t]

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Use AllDependentVariables with the resource function EulerEquations to compute the corresponding Euler-Lagrange equations of motion for the double pendulum:

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$Ldp = 1/6 m l^2 (\[Theta]2'[t]^2 + 4 \[Theta]1'[t]^2 + 3 \[Theta]1'[t]*\[Theta]2'[t]* Cos[\[Theta]1[t] - \[Theta]2[t]]) + 1/2 m g l (3 Cos[\[Theta]1[t]] + Cos[\[Theta]2[t]]);$