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We have a bead sliding along a smooth wire in the shape of a cycloid with equations, $$ x=a(\theta-\sin\theta),\space y=a(1+\cos\theta), $$where $0\leq\theta\leq2\pi$. To find the number of coordinates required to describe the motion of the particle (call it $N$) we can use $3n-c=N$ where $n$ is the number of particles and $c$ is the number of constraints on the particles. I know that $N=1$ so we must have $c=2$. I also know that the first constraint is, $$\phi_1(\textbf{x},t) = \textbf{x}\cdot\textbf{n},$$where $\textbf{n}$ is perpendicular to the plane of the wire.
My question is what would be the function $\phi_2(\textbf{x},t)$ to constrain the bead to the wire?

K.defaoite
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Kian31
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  • In which context do you need this constraint? In a Lagrangian for instance? – Cesareo Mar 08 '24 at 11:31
  • @Cesareo It’s to motivate and justify the use of one coordinate ($\theta$) to find the Euler-Lagrange Equation. – Kian31 Mar 08 '24 at 13:08
  • Here, the configuration space dimension is $2$ as $(x,y)$, with one constraint, because we can determine the holonomic dependency $y = f(x)$ so the movement is one dimensional $(2-1)=1$. Along the constraint. – Cesareo Mar 08 '24 at 13:58

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