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DependentVariableQ (1.1.1) current version: 1.2.0 »

Source Notebook

Determine whether an expression is a dependent variable

Contributed by: E. Chan-López & Jorge Luis Ramos Castellano

ResourceFunction["DependentVariableQ"][expr,vars]

gives True if expr depends on one or more variables vars, and gives False otherwise.

Details

ResourceFunction["DependentVariableQ"] threads over lists.

Examples

Basic Examples (1)

Use DependentVariableQ to identify a variable that depends on a single variable:

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Scope (3)

Use DependentVariableQ with a list of variables that depend on a single variable:

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Use DependentVariableQ with a variable that depends on two variables:

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These are not dependent variables:

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$ResourceFunction["DependentVariableQ", ResourceVersion->"1.1.1", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{Tan[\[Theta]], Cos[\[Theta]]}, \[Theta]]$

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Applications (8)

Define a simple function to identify all dependent variables of a single variable for a given Lagrangian and Euler-Lagrange equations:

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$DependentVars[g_, indepvar_] := Map[If[ResourceFunction["DependentVariableQ", ResourceVersion->"1.1.1", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][#, indepvar] === True, #, Nothing] &, Sort[DeleteDuplicates@ ReplaceAll[Derivative[_][x_][_] :> x[indepvar]][ DeleteDuplicates[ Flatten[Which[Depth[#] == 1 \[Or] Depth[#] == 2, #, Depth[#] >= 3, Level[#, {Length[Depth[#]] - 2}]] & /@ Level[g, {1}]]]]]]$

Lagrangian for the double pendulum:

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$L1 = 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]]);$

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Lagrangian for the spherical pendulum:

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$L2 = 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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Lagrangian for the PUMA-Like Robot:

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$L3 = 1/2 m2 (x2'[t]^2 + y2'[t]^2 + z2'[t]^2) + 1/2 m3 (x3'[t]^2 + y3'[t]^2 + z3'[t]^2) + 1/2 (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) \[Psi]1'[ t]^2 + 1/2 A2 \[Theta]2'[t]^2 + 1/2 A3 \[Theta]3'[t]^2 - g (m2 z2 + m3 z3);$

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Use DependentVars with the resource function EulerEquations to compute the corresponding Euler-Lagrange equations of motion for the Lagrangians L1, L2 and L3:

Euler-Lagrange equations for the double pendulum:

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Euler-Lagrange equations for the spherical pendulum:

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Euler-Lagrange equations for the PUMA-Like Robot:

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Properties and Relations (4)

Use DependentVariableQ with the resource function SolutionRulesToFunctions to convert solution rules to function rules in a given list containing rules whose left-hand side don't match with a variable that depends on other variables:

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Map[If[Head[#] === Rule && ResourceFunction["DependentVariableQ", ResourceVersion->"1.1.1", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][#[[1]], t] === True, ResourceFunction["SolutionRulesToFunctions"][#], Nothing] &, {m, s,
q, y[t] -> a x[t], z[t] -> c b w[t]}]

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Use DependentVariableQ with the resource function SolutionRulesToFunctions on a more complicated list:

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$Map[If[Head[#] === Rule && (ResourceFunction["DependentVariableQ", ResourceVersion->"1.1.1", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][#[[1]], t] === True \[Or] ResourceFunction["DependentVariableQ", ResourceVersion->"1.1.1", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][#[[1]], {x, t}] === True), ResourceFunction["SolutionRulesToFunctions"][#], Nothing] &, {1, Cos[t], m, s, q, y[t] -> a x[t], sol[x, t] -> Sinc[x - t]}]$