Add GetFreeVariables() overload to Expression (#22145)

We have Expression::GetVariables() and Formula::GetFreeVariables().
In #22091, we could see that this difference (which is mostly
unnecessary) made the downstream code unnecessarily clunky. This
simple overload resolves it.

bindings/pydrake/symbolic/symbolic_py_monolith.cc

@@ -354,6 +354,8 @@ void DefineSymbolicMonolith(py::module m) {
           py::arg("env"), doc.Expression.EvaluatePartial.doc)
       .def("GetVariables", &Expression::GetVariables,
           doc.Expression.GetVariables.doc)
+      .def("GetFreeVariables", &Expression::GetFreeVariables,
+          doc.Expression.GetFreeVariables.doc)
       .def(
           "Substitute",
           [](const Expression& self, const Variable& var, const Expression& e) {

bindings/pydrake/symbolic/test/symbolic_test.py

@@ -691,6 +691,8 @@ def test_get_variables(self):
         vars = e_x.GetVariables()
         self.assertEqual(len(vars), 1)
         self.assertTrue(list(vars)[0].EqualTo(x))
+        vars2 = e_x.GetFreeVariables()
+        self.assertTrue(vars2.EqualTo(vars))
 
     def test_get_variable_vector(self):
         vars_ = sym.GetVariableVector([e_x, e_y])

common/symbolic/expression/expression.h

@@ -193,6 +193,10 @@ class Expression {
   /** Collects variables in expression. */
   [[nodiscard]] Variables GetVariables() const;
 
+  /** Same as GetVariables(); we provide this overload for compatibility with
+   * Formula. */
+  [[nodiscard]] Variables GetFreeVariables() const { return GetVariables(); }
+
   /** Checks structural equality.
    *
    * Two expressions e1 and e2 are structurally equal when they have the same

common/symbolic/expression/test/expression_test.cc

@@ -1925,13 +1925,14 @@ TEST_F(SymbolicExpressionTest, GetVariables) {
   EXPECT_FALSE(vars1.include(var_z_));
   EXPECT_EQ(vars1.size(), 2u);
 
-  const Variables vars2{(x_ * x_ * z_ - y_ * abs(x_) * log(x_ + y_) + cosh(x_) +
-                         cosh(y_) + atan2(x_, y_))
-                            .GetVariables()};
+  const Expression e = x_ * x_ * z_ - y_ * abs(x_) * log(x_ + y_) + cosh(x_) +
+                       cosh(y_) + atan2(x_, y_);
+  const Variables vars2{e.GetVariables()};
   EXPECT_TRUE(vars2.include(var_x_));
   EXPECT_TRUE(vars2.include(var_y_));
   EXPECT_TRUE(vars2.include(var_z_));
   EXPECT_EQ(vars2.size(), 3u);
+  EXPECT_TRUE(e.GetFreeVariables() == vars2);
 }
 
 TEST_F(SymbolicExpressionTest, Swap) {