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In D'Alembert principle forces are classified into constraint and applied forces? Is this classification different from internal-external forces?

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Yes, they are different.

One must define what the system under study is. Usually, it consists of a number of sub-systems. A marble, for example, has as sub-systems many many atoms. Two point masses connected by a massless spring has two sub-systems. Internal forces are forces between sub-systems: the interatomic forces in a marble, and the spring force for the two point masses. External forces would be the force applied by agents external to the system: the earth in the case of gravity, my thumb in the case of a marble.

For the applied force, the agent and the functional form of the force (or potential) is specified. For the constraint, the agent might be known (a ramp, perhaps) but the functional form is not. Typically we limit the path or parameter space of the system in some way. We simply say that the constraint agent can provide whatever force is necessary to achieve the constrained motion.

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  • $\begingroup$ Constraining forces need not be external. Consider two masses connected by a rigid rod. The constraining forces will be mutually opposite with the net force being zero. $\endgroup$
    – Jonas Greitemann
    Commented Jun 19, 2014 at 15:41
  • $\begingroup$ @Jonas True enough. I left myself some wiggle room in the wording, but I think I'll delete that sentence. $\endgroup$
    – garyp
    Commented Jun 19, 2014 at 15:44
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Yes. External forces do not have an opposite counter force to another object within the system under consideration. Constraining forces limit motional degrees of freedom.

E.g. take gravity. You are pull down to Earth with force $mg$ but at the same time Earth experiences a pull towards you with an equal but opposite force. This is what is known as an internal force. But often, Earth is not included into the description of the system b/c its so massive and the force due to you will barely make move. So one make an abstraction by considering the Earth surface fixed, effectively taking it out of the system and subjecting everything to a downward force. This is an example for an external force. But its not a constraining force. You see that the distinction between external and internal forces depends on the choice of the system you consider.

A constraining force is a force which limits the degrees of freedom of the motion of an object. E.g. a pendulum with a mass attached at the end of a rigid rod can only rotate around one axis. So, the motional freedom is down from 3 directions (x,y,z) to one, the angle parametrizing the the position on the circle. In Newtonian mechanics, we would still describe the pendulum using 3D force, position and velocity vectors but the constraining forces will cancel any forces which are not tangential to the circle the pendulum mass is constrained to. D'Alembert's principle states that those constraining force are orthogonal to the surface along which the motion can occur. In Lagrangian mechanics, one switches to generalized coordinates, in this case the angle, i.e. one implements the constraints by limiting the motional degrees of freedom right away.

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