I'm looking for a way to extract a list of variables from an expression, for example with an input like:

Leff= (mc dm^2 + mc/12*(h^2 + 3 R^2) + ma da^2 + ma/12 La^2)/(mc dm + ma da)

I want this output:

{mc, dm, ma, da, La, h, R}.

The built-in Mathematica function Variables can do this, but it doesn't work with more complex expressions containing trascendental functions. Any help would be very appreciated.

  • $\begingroup$ Wouldn't it be more natural to use { c, d, a, b, L, h, R} instead of unreasonable and awful {mc, dm, ma, da, La, h, R} ? $\endgroup$
    – Artes
    Mar 13, 2013 at 20:24
  • 6
    $\begingroup$ @Artes Mnemonics can often be helpful, as well as using variables whose names are as close as possible to a formula as printed in a textbook or paper. Let's blame the paper authors and not the coders for bad variable names! $\endgroup$
    – whuber
    Mar 13, 2013 at 20:41
  • $\begingroup$ Related: (15153) $\endgroup$
    – Mr.Wizard
    Mar 13, 2013 at 23:24

5 Answers 5


Assuming you don't have any built-in symbols in that list, you could simply do:

DeleteDuplicates@Cases[Leff, _Symbol, Infinity]
(* {da, ma, dm, mc, La, h, R} *)

If you do have symbols from built-in contexts or packages, you can simply pick out only those that are in the Global` context with:

With[{globalQ = Context@# === "Global`" &},
    DeleteDuplicates@Cases[Leff, _Symbol?globalQ, Infinity]

If you have a different default working context (e.g. local to notebook/cell or in a package), change the pattern test to the following, instead of globalQ:

currentContextQ = Context@# === $Context &
  • 1
    $\begingroup$ Thank you very much, this is exactly what I was looking for. $\endgroup$
    – John
    Mar 13, 2013 at 20:32

Using an undocumented function:

Reduce`FreeVariables[(mc dm^2 + mc/12*(h^2 + 3 R^2) + ma da^2 + ma/12 La^2)/
                     (mc dm + ma da)]
   {da, dm, h, La, ma, mc, R}

The below code for getAllVariables was lifted without attribution from some StackOverflow post.

headlist = {Or, And, Equal, Unequal, Less, LessEqual, Greater, 
   GreaterEqual, Inequality};

getAllVariables[f_?NumericQ] := Sequence[]
getAllVariables[{}] := Sequence[]
getAllVariables[t_] /; MemberQ[headlist, t] := Sequence[]

getAllVariables[ll_List] := 
 Flatten[Union[Map[getAllVariables[#] &, ll]]]

getAllVariables[Derivative[n_Integer][f_][arg__]] := 

getAllVariables[f_Symbol[arg__]] := 
  If[MemberQ[Attributes[f], NumericFunction] || MemberQ[headlist, f], 
   fvars = getAllVariables[{arg}],(*else*)fvars = f[arg]];

getAllVariables[other_] := other


Leff = (mc dm^2 + mc/12*(h^2 + 3 R^2) + ma da^2 + ma/12 La^2)/(mc dm +
     ma da)

(* Out[254]= {da, ma, dm, mc, ma, da, ma, La, mc, dm, mc, h, R} *)
  • $\begingroup$ Daniel, since questions are generally not being migrated any longer and since these valuable posts ultimately fall outside our control, why don't you repost your method as a full answer? $\endgroup$
    – Mr.Wizard
    Mar 16, 2013 at 14:15
  • 4
    $\begingroup$ @Mr.Wizard Done. (But if you were a real wizard, couldn't you just make it magically happen?) $\endgroup$ Mar 16, 2013 at 20:44

If you have even more complicated expressions, you might want to use Heads -> True.

expr = {f, Subscript[g, i], h[i[j[a, b]]], s'[t] == u[t] + v[t]};

Union @ Cases[expr, Except[__Symbol?(Context @ # === "System`" &), _Symbol], {1, ∞},
              Heads -> True]
{a, b, f, g, h, i, j, s, t, u, v}

Without checking heads:

Union @ Cases[expr, Except[__Symbol?(Context @ # === "System`" &), _Symbol], {1, ∞}]
{a, b, f, g, i, t}
  • 2
    $\begingroup$ This is the method I would use, expect perhaps as Union @ Cases[Leff, s_Symbol /; Context[s] =!= "System`", -1] $\endgroup$
    – Mr.Wizard
    Mar 13, 2013 at 23:11
  • $\begingroup$ @Mr.Wizard I've tried to provide code that is sufficiently different from rm-rf-s version not to give room for any comments on plagiarism :) $\endgroup$ Mar 14, 2013 at 7:46
  • $\begingroup$ Thank you, the expressions I usually work with are not so complicated but I might need this sometimes. $\endgroup$
    – John
    Mar 14, 2013 at 15:19
expr = (mc dm^2 + mc/12*(h^2 + 3 R^2) + ma da^2 + ma/12 La^2)/(mc dm + ma da); 
Level[expr, {-1}] /. x_?NumericQ :> (## &[]) // DeleteDuplicates
(* {da,ma,dm,mc,La,h,R} *)

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