I was brushing up on my DOF concepts before moving on to Lagrangian mechanics. One of my professors told me that a spring is not considered a constraint but his explanation was not satisfactory in my POV. EDIT: A Mechanical system’s Degree of Freedom (DOF) is the number of independent characteristics that describe its configuration or state. The restrictions on the motion of a system are called constraints.
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$\begingroup$ What's DOF mean? And what do you mean by "constraint"? $\endgroup$– Bob DCommented Jul 24 at 12:04
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$\begingroup$ @BobD I have edited my question. Hope it adds more context $\endgroup$– Harshitha SridharCommented Jul 24 at 12:13
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$\begingroup$ Also it would have helped context by at least summarizing the description your professor gave that you felt was unsatisfactory...perhaps we could have shed some light on it. $\endgroup$– TriatticusCommented Jul 24 at 17:53
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$\begingroup$ A compression spring for example can deflect along its length, bend and twist as a whole when bent by axial forces and twisting moments. There appear 3 DOF in an ode set up in static/dynamic situations. $\endgroup$– NarasimhamCommented Jul 25 at 16:27
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3 Answers
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Let the vector $\vec{x}(t)$ be the dynamical variable whose motion you're trying to solve for. Since it is a vector, we can say there are $3$ degrees of freedom, or DOFs. (Actually if you go further you will find there are different ways of counting DOFs in different schemes, but let's say there are $3$ here for simplicity).
A typical equation $x$ might obey that we need to solve is $$ m \ddot{\vec{x}} = -V'(x) $$ The motion of $x(t)$ is obtained by solving this dynamical equation. The only information we need to specify are the initial conditions $\vec{x}(0)$ and $\dot{\vec{x}(0)}$, and then we can solve this equation and obtain the motion. The variable $\vec{x}(t)$ is free to take on any values inside of $\mathbb{R}^3$; solving the equation of motion will tell us what values $\vec{x}(t)$ takes. Note that the equation for a mass on a spring takes the form of the above equation, with $V(x)=\frac{1}{2}x^2$.
Without getting too deep into all the different kinds of constraints there are, a constraint is an additional requirement beyond the equation of motion that $\vec{x}$ must satisfy. A typical example might be that we restrict $\vec{x}$ to live on a surface, such as a sphere. Then we might add an equation $$ |\vec{x}| = R $$ Then our goal is to solve the equation of motion $m\ddot{x}=-V'(x)+F_{\rm constraint}$ subject to the constraint $|\vec{x}|=R$. The constraint is an additional equation, beyond the equation of motion, that restricts the solution to some smaller domain than $\mathbb{R}^3$. Note that I've had to add a constraint force $F_{\rm constraint}$ to Newton's law -- Newton's laws without the constraint may "want" the particle to leave the surface of the sphere, so to make sure the constraint is satisfied, an additional force must be applied to keep the particle on the surface. As a simple example, gravity wants to pull you to the center of the earth, but you can imagine a constraint that you must live on the surface. The constraint force would then be the normal force keeping you from falling to the center.
To summarize, a spring corresponds to a potential and can be inserted into a dynamical equation you solve for $\vec{x}(t)$. It is not a constraint, which is an additional requirement that restricts the domain of solutions to the dynamical equation.
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The answer depends on what one means by "constraint." Consider that in control theory, one derives the Hamiltonian by treating "the system must obey all physical laws" as a "constraint" and manipulating the equations from there. That is certainly one valid meaning for the word "constraint." However, when we talk about "Degrees of Freedom," we are typically talking about situations where we can analytically reduce the configuration space to a smaller dimensionality.
The classic example being that a pendulum swinging in 2 dimensions (x and y) cannot possibly reach all possible states in that 2 dimensional space, and that in fact there exists a 1 dimensional space (e.g. theta) which captures all of those possible states. When speaking of Lagrangian Mechanics, most instructive documents I know of are indeed thinking of this second meaning. It's the basis for why expressing physics in a form amenable to curvilinear coordinates (as Lagrangian Mechanics does) is a desirable thing.
Usually a spring is not assumed to restrict the space in such a way. Intuitively, if we replace said pendulum with a spring, it's clear to see that we can theoretically reach a 2 dimensional space by mentally thinking about how that spring could bounce up and down over time while swinging side to side.
Devil is in the details: it may be that a particular spring is tuned such that you still have a 1 dimensional space, it's just more complex than the pendulum example (part of your task may be to prove this). It's also possible that the spring is so stiff that treating the spring as a constraint is a reasonable assumption for your particular application (remember: all models are wrong; some are useful)
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An ideal spring with an infinite (finite) spring constant $k$ is (is not) a constraint as it does (does not) restrict the configuration space/the DOF, respectively.