Analytical Mechanics: Regularity Conditions on Constraint Surface

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In analytical mechanics, regularity conditions on the constraint surface G = 0 are crucial because they ensure that the equations of motion yield a unique solution for the particle's trajectory. If the gradient of G is zero at a point, such as (0,0) in the example provided, the particle can move in multiple directions, leading to ambiguity in its motion. This lack of a unique solution violates the principle that a particle's motion should be determinable from its initial position and velocity. The discussion also touches on the relationship between the gradient and the particle's trajectory, suggesting that the particle's velocity is influenced by the gradient of G. Understanding these concepts is essential for correctly applying constraints in mechanical systems.
apec45
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Hi,
In my course in analytical mechanics, it is said that for a system of n particles subjected to r constraint equations, it is necessary to impose regularity conditions on the constraint surface defined by G = 0 where G is a function of the position of the position of the particles and time, the condition is that the gradient of G is non-zero on the surface G = 0.

I don't understand why we're asking this?

Thanks for your help (and sorry if i made mistakes)
 
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apec45 said:
don't understand why we're asking this?
because this condition guarantees that the equations of motion will be correct

For example if you have a particle and the constraint ##G=x^2-y^6=0## on the plane, then the particle will not know where to move from ##(0,0)##: it can move along the curve ##x=y^3## or along the curve ##x=-y^3##. In this example ##\nabla G\mid_{(0,0)}=0##
 
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Thanks you for the answer! but i don't understand it very well :(

First, I don't see the link between "the gradiant is zero somewhere" and "there is several ways to move the particule somewhere" (I am ok with your example but i don't see the link).

Then, i don't know what's the problem to have several ways to move the particle, for example if there was no constraints (G = 0 = 0 if i can say that), there would be infinitely many ways to move the particule (anywhere on the plane here) but i don't see the problem in this case, in fact i don't see the link with "the equation of movement has exactly one solution"

Thank you for your help!
 
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A motion of the particle must be determined uniquely by its initial position and velocity. In my example for the initial position take (0,0) and for the initial velocity take ##\boldsymbol {e}_y##. So where will the particle move then?
 
Oh okey, i got it! Thank you so much! We don't know where the particule will go (x = y³ or x = -y³).

And could you give me a little bit explanation about the link with the gradiant? :) I really don't see the link with that, or maybe it is that : if G(x, y) = 0 is the equation of the surface then the particule will have, on a point (x, y) of the surface, a velocity proportional to the gradiant of G on (x, y) ? (i mean if we consider a gradiant like a vector field then if we let a particule follow the vector field, we can see its trajectory)

ps: according to me if the gradiant on a point is zero then the particule will stay there and keep that position, it doesn't seem a problem, or am i wrong?
 
apec45 said:
And could you give me a little bit explanation about the link with the gradiant?
Solve the following problem. A particle of mass m slides without friction on a surface ##\{f(x,y,z)=0\}##. The function ##f## is given. For simplicity assume that there is no other forces except the force of normal reaction ##\boldsymbol{N}.## Find $$\boldsymbol {N}=\boldsymbol {N}(\boldsymbol {r},\boldsymbol {\dot r}).$$
Here ##\boldsymbol r=x\boldsymbol {e}_x+ y\boldsymbol {e}_y+z\boldsymbol {e}_z## is the radius-vector of the particle in the standard inertial frame ##Oxyz##
 
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Uhm, i only see that ##N = m . a## (by Newton's second law) and ##f(x, y, z) = 0## (constraint)
 
I also see that if i do that: ##N.t = ma.t## with## t ## tangent to the surface, i get ##a*t = 0##
 
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