# Lagrangian mechanics

I, beginner to Lagrangian mechanics, was reading Hand and Finch and got stuck with a concept.it was the development to lagrangian formulation by treating a problem, first in newtonian way, then using virtual work consideration and finally, the Lagrangian way,the last is still beyond my read part.However, the problem was a small bock sliding downwards along the surface of a smooth wedge, free to move on a frictionless floor...when calculating virtual work of the wedge, HF says that 'WE DO NOT HAVE TO TAKE INTO ACCOUNT THE CONSTARINT FORCES ...,BECAUSE THE CONSTRAINT FORCES ALWAYS ACT IN A DIRECTION PERPENDICULAR TO THE POSSIBLE DISPLACEMENTS';they subsequently calculate using only gravity force.As you see the constraint force, that one which is normal reaction of the floor, does no work. But the force of reaction due to the block resting on the wedge, if it is a constraint force, why it has no contribution in the virtual workof the wedge?Even if it is not a constraint force to the wedge, then also it should have contribution to the virtual work...Where I am going wrong?

constraint forces are usually internal forces, hence they do no net work. think of the constraint in this particular problem as conservation of momentum, the momentum in the horizontal direction must be constant.

I agree with you that constraint force, here atleast, is internal force.And it is also true that internal forces do not work on a system as a whole. But what do you mmean by latter statements?

well in the particular problem you described, the wedge problem, the momentum in the horizontal direction is conserved, since there is no horizontal force on the total system.

so what?What has that to do with the virtual work?

I think the other constraint force is the wedge against the block.

Forces of constraint can do work if they're time dependent. This is a rather important point.

If you think about constraint... Well, what is a constraint, really? a constraint is a function relating a constant to variables. for example, assuming time independence:
$$f(q_1,q_2)=C$$
the constraint force is the gradient of f, which is always orthogonal to the level surface described by the constraint, hence,
$$\vec{\nabla} f\cdot{d\vec{r}}=0$$

you can think of constraint force as a force that keeps a system in the level "surface" described by the constraint, so that the constraint force has to be orthogonal to that "surface" thus the work done by the constraint force is zero.

just a note: in my book, the definition of constraint force is implicitly stated as "the force that does no work"...it is just a stated fact.

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If you think about constraint... Well, what is a constraint, really? a constraint is a function relating a constant to variables. for example, assuming time independence:
$$f(q_1,q_2)=C$$
the constraint force is the gradient of f, which is always orthogonal to the level surface described by the constraint, hence,
$$\vec{\nabla} f\cdot{d\vec{r}}=0$$

you can think of constraint force as a force that keeps a system in the level "surface" described by the constraint, so that the constraint force has to be orthogonal to that "surface" thus the work done by the constraint force is zero.

just a note: in my book, the definition of constraint force is implicitly stated as "the force that does no work"...it is just a stated fact.

But as I said earlier, there are some constraints that do actually do work. Look at the bead on a loop problem. You can view it as being the constraint that $$\theta = \omega t + \theta_0$$, but it's most definitely doing work. Energy is not conserved in that case. Thankfully, it's an holonomic constraint, but it is adding energy to the system none the less.

Constraints arise from forces that cannot be classified easily as potentials, but which don't necessarily act except in a way to force the particle to behave within a certain set of parameters. If I have a mass on a track and constrain it to that track, there's a normal force keeping it on the track, but it's much easier to simply state that the particle has to be along this track than to characterize the normal forces, and this is one of the great power of lagrangian mechanics over straight Newtonian mechanics.