I'm starting to learn about Degrees of freedom, and the idea of 'constrained motion' seems strange to me, surely any particle with a predefined path is 'constrained' in its motion, We also had described to us that if its limited to a plane it will have lower degrees of freedom, this makes sense in so much as a 3D path may not live on a plane so it makes sense that there is less freedom for motion, however they then said if its limited to a linear path, it will also be constrained, surely any particle on a particular path is limited to a linear path? How is it more constrained in this scenario? How can these values be 'parameters' when most of them are related and dependent on each other (usually by their relation to time)
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When we speak of degrees of freedom for a particle, we essentially mean the number of spatial coordinates we need to describe the motion of the particle.
To describe the position of a free particle, we need three such coordinates; $x$, $y$ and $z$ if we’re using a Cartesian coordinate system. This means that in order to describe the motion of the particle, we need to know its $x$, $y$ and $z$ coordinates as functions of time, and a free particle therefore has three degrees of freedom. You could think of this as the default case.
Now, when we introduce a constraint, we essentially create some relation between two or several spatial coordinates (or force some coordinate to always remain constant). Each such relation, or constraint, effectively reduces the number of degrees of freedom by one.
A simple example would be a ball rolling on a table. The $z$-coordinate of the ball remains constant and equal to the height of the table for all $t$, and for the sake of describing the motion of the ball, we don't even have to know what it is. We only need to know $x(t)$ and $y(t)$ to describe the motion, and we say that the ball has two degrees of freedom.
Now say we place a straight pipe on the table and let the ball roll back and forth inside the pipe. For simplicity, we align the $x$-axis with the pipe. We now only need to know $x(t)$ to describe the ball's motion, and it therefore has one degree of freedom.
In general, constraints are usually a bit more complex than this. We could imagine that the table is not flat and that $z$ is some function of $x$ and $y$; $z = f(x,y)$. But this still means that if we can find $x(t)$ and $y(t)$, we automatically also know $z(t)$. So the idea remains the same; some relation between the coordinates exists, we call this a constraint, and it reduces the number of degrees of freedom by one.
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$\begingroup$ But x(t), y(t) and z(t) can all be related to each other due to being a function of time? $\endgroup$ Commented Apr 11, 2022 at 12:25
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$\begingroup$ Does it have to be explicit relations between all of the co-ordinates? e.g. we know that $z=f(x,y)$, and not something like $x(t)=g(y(t))=h(z(t))$ $\endgroup$ Commented Apr 11, 2022 at 12:31
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$\begingroup$ Is it more that even though I know $x(t)$ and $y(t)$ are functions of the same parameter and can be related to each other, in the general scenario, knowing $x(t)$ does not mean I know the function that $y(t)$ can be certainly if given the scenario, i.e. even though I might be able to define a relation between them, it's only because I know them both, where as a constraint is an outside influence that allows me to know $y(t)$ when I know $x(t)$ in all similar cases? So in unconstrained motion, x(t) and y(t) have a relation but its still 3 degrees of freedom? $\endgroup$ Commented Apr 11, 2022 at 12:42
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$\begingroup$ @user1007028 The idea is that degrees of freedom are useful because they tell us how many equations we need to write down and solve in order to completely describe the time-evolution of a physical system. Sure, $x$, $y$ and $z$ are all related by $t$, and if we figure out these relations for a free particle, we might in theory be able to write something like $x = f(y)$, $y = g(z)$, $z = h(t)$ but there wouldn’t really be much point since we still need at minimum three equations to properly describe the time-evolution of the system. $\endgroup$ Commented Apr 11, 2022 at 15:07
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$\begingroup$ So, the idea then is if there's sort of some almost external thing that is constraining $x,y,t$ and defining more about them just than just the relation that might just happen between them, depending on what the motion actually is being constrained if you say before a constraint is applied such that we know a certain relation has to be true for this motion, without knowing anything else about the system at all? So if I had three different type of motions being constrained in the same way, I'd know this relation to be true, regardless of what the motion actually is? Almost like a generalization $\endgroup$ Commented Apr 11, 2022 at 16:13