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I was doing a question which was to find the number of generalized coordinates needed to describe a particle with the motion:

$x(t)=2a\sin(\omega t) $

$y(t)=a\cos(2\omega t)$

So I solved it and found the equation of motion to be:

$(y-a)=\frac{-x^2}{2a}$

So what I am confused about is what can be called a constraint equation, here can the equation of motion can be called a constraint equation or not? According to me the equation of motion constrains the given particle to move in just the given path, but at the same time, I might've found the equation of motion due to the constraint. Please help.

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    $\begingroup$ When physicists say "the equation of motion" they usually mean a differential equation (or a system of equations) that you can solve using a set of given initial conditions. Your equation is not a differential equation (it relates the coordinates algebraically) so it's a constraint equation, not an equation of motion. $\endgroup$
    – Michael Seifert
    Commented Jan 27 at 14:35
  • $\begingroup$ oh, okay that makes sense. But if I had to write the equation of motion, what would that be? Are the two things given like x=2asin(wt) and the y(t) the equation of motion? $\endgroup$
    – Kutubkhan Bhatiya
    Commented Jan 27 at 15:25

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Constraint equation usually relates dependent parameters, how they are connected to each other, i.e. having variables $p,q$,- constraint equation would be expression $p^n-q^m$ or $~{p^n}/{q^m}$. So in very technical terms $y-a$ is not a constraint equation, because it's just a subtraction of amplitude.

One form of constraint equation in your case could be for example : $$ \tag 1 \frac{x^2}{y} = \frac{4a}{\cot^2(\omega t)-1} $$

Equations of motion are system of equations $x(t),~y(t),~z(t)$. In mathematics world these are called parametric equations, but we physicists usually omit word "parametric" from our vocabulary, I assume mostly because majority of equations in Kinematics are parametric ones.

Hope that helps.

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  • $\begingroup$ So, the equation that I wrote is not a constraint equation then? Is it because we are not relating x and y with the parameter here that is t? $\endgroup$
    – Kutubkhan Bhatiya
    Commented Jan 27 at 18:04
  • $\begingroup$ Because if instead of the above question, we took something like x=asin(wt) and y=acos(wt), then if I try to solve it in polar coordinates wouldn't I get two constraint equations one would be r=a and the other would be theta in terms of t, so aren't these two separate constraint equation, so when I try to find the number of generalized coordinates, it would come zero, but that is wrong so one of them is not a constraint equation then, which one would it be? $\endgroup$
    – Kutubkhan Bhatiya
    Commented Jan 27 at 18:10
  • $\begingroup$ No, if you convert it into polar coordinate system like, $r(t), \theta (t) $ it will still be parametric equation system, because both parameters depends on $t$. r can also depend on time, $r=r(t)$, imagine spiral. Constraint equation relates both parameters together, so that in terms of one you can find another, but like I've said it should be expressed as subtraction or ratio of dependent variables. $\endgroup$
    – Agnius Vasiliauskas
    Commented Jan 27 at 18:49
  • $\begingroup$ Yeah, that makes sense. $\endgroup$
    – Kutubkhan Bhatiya
    Commented Jan 28 at 6:52
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Isn't the number of generalized coordinates simply 1? ($t$). If you give me $t$, I can specify the location of the particle (because the equations $x(t)$ and $y(t)$ are given).

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  • $\begingroup$ Yeah, I do feel the same, but my question is not about how many generalized coordinates there are, it is mainly about what is the constraint equation, is it the equation of motion, or is it something else which I might've used to reach the equation of motion. $\endgroup$
    – Kutubkhan Bhatiya
    Commented Jan 26 at 18:42
  • $\begingroup$ I now understand, that does sounds tough. I wish I had a better answer then, I am also confused what "the constraint equation" there means $\endgroup$
    – Jack
    Commented Jan 26 at 20:35

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