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Let's consider a system described by this simple equation:

$x^{2}+y^{2}=5$

this represents a holonomic constraint. Suppose I would like to reduce the degrees of freedom of this system, I should be able to do it since is described by a holonomic constraint. However, both $x^{2}$ and $y^{2}$ are not invertible. So how can I write one of them as a function of the other?

BioPhysicist
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AleWolf
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    Can you think of a better coordinate system than rectangular coordinates? – Qmechanic Nov 24 '19 at 17:22
  • Yes, i've understood. However what i don't understand is which are the advantages of expressing a system as an equation $f(q_{1},...,q_{n})=0$. Can you explain to me ? – AleWolf Nov 24 '19 at 17:52
  • In practice, the physical system under consideration may imply the holonomic constraint $f(q,t)\approx 0$ directly rather than some other version. – Qmechanic Nov 24 '19 at 18:02
  • Why, intuitively, in cartesian coordinates isn't immediate to see that there is only 1 degree of freedom, whereas in polar coordinates you immediately see it ? – AleWolf Nov 24 '19 at 20:52
  • Another thing i have not understood. Why if my system is described by $f(q_{1},...,q_{n})=0$ i can ALWAYS represent one variable as a function of the other ? – AleWolf Nov 24 '19 at 21:11

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