For nonholonomic systems, commuting the variation and time derivative forces the Lagrangian derivatives of the constraints to lie in the span of the constraint gradients.
(2011) The non–holonomic double pendulum, an example of non-linear non-holonomic system,Regular and Chaotic Dynamics,1 n
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On the transpositional relation for nonholonomic systems
For nonholonomic systems, commuting the variation and time derivative forces the Lagrangian derivatives of the constraints to lie in the span of the constraint gradients.