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I'm learning about Lagrange equations and was wondering if the lagrange equations would hold with moving constraints. I've searched a lot on the internet, but never came across my exact question. I know that in order to use lagrange equation the system needs to be holonomic. Although I find the definition of a holonomic system somewhat hard to grasp it goes as follows: If we can write the constraints as follows $$\vec{r}_{a}=\vec{r}_{a}\left(q_{1}, \ldots, q_{K}\right)$$ we are dealing with a holonomic system and if not we're dealing with a non-holonomic system.

Knowing this I would answer on my question that the lagrange equations do not hold, because my interpretation of a moving constraint is that it is time dependent and does not fit the definition of a holonomic system. I like to convince myself of my answer but I'm not quite convinced.

If someone could explain to me why I'm wrong or maybe why my argument is right it would be very helpful!

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    $\begingroup$ Could you put in an example of what you mean by a moving constraint? $\endgroup$
    – Dan
    Jan 19, 2022 at 14:26
  • $\begingroup$ Hi TheCreator. Welcome to Phys.SE. For what it's worth, holonomic constraints allow explicit time dependence, cf. e.g. Wikipedia. Where did you hear otherwise? $\endgroup$
    – Qmechanic
    Jan 19, 2022 at 14:46
  • $\begingroup$ I have included an example in an old post of mine here: physics.stackexchange.com/q/652126 $\endgroup$ Jan 19, 2022 at 16:47

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Everything works well with constraints which depend on time explicitly.

The definition of holonomic system of constraints is that, for $N$ material points represented in Cartesian coordinates in the rest space of a reference frame (the definition does not depend on the choice of the reference frame) a set of $c< 3N$ conditions must hold $$f_j (t, \vec{x}_1, \ldots, \vec{x}_N)=0\quad j=1,\ldots,c\:, \tag{1} $$ where the functions f_j take values in $\mathbb{R}$ are $C^k$ with $k>1$ and, where (1) is valid these functions are functionally independent for every $t$. In other words, the Jacobian matrix of derivatives $\frac{\partial f_j}{ \partial y_l}$ where $y_1,\ldots, y_{3N}$ are the components of the vectors $\vec{x}_1,\ldots, \vec{x}_N$, has rank $c$ when (1) holds.

In this case, locally, it is possible to describe every $\vec{x}_i$ as a function of $t$ and $n= 3N-c$ free components, denoted by $q^1,\ldots, q^n$ among the $y_r$. Different choices are also possible but, locally in space and time, the allowed configurations are defined by $n$ free coordinates.

If the constraints satisfy the physical requirement of ideal constraints regarding the behavior of the reactive forces, the Newton equations are equivalent to the usual Euler-Lagrange equations
written in terms of a curve in the space of the free coordinates.

The requirement of ideal constraint generalize the one of frictionless constraint including some further, physically relevant, possibilities as the rigidity constraint.

You see that there is no restriction on the time dependence of the functions $f_j$.

As an example, think of a curve $\Gamma= \Gamma(t, s) \in \mathbb{R}^3$, in the rest space of a reference frame, whose shape changes in time $t$ and where $s\in \mathbb{R}$ is a regular parameter along the curve. For instance $$\Gamma(t,x) = (r+ ct^2) (\cos s {\bf e}_x + \sin s {\bf e}_y)\:.$$ where $r, c\in \mathbb{R}\setminus \{0\}$ are given constants.

This is a circle whose radius $R(t) = (r+ ct^2)$ depends on time. This space can be obtained by imposing the constraints $$f_1(t,x,y,z) := z \equiv 0$$ and $$f_2(t,x,y,z) := (r+ ct^2)^2 - x^2 -y^2 \equiv 0\:.$$ It is not difficult to see the the associated Jacobian matrix has rank 2 where the two constrains are valid. Locally one can use the coordinate $x$ or the coordinate $y$ to describe the configuration space, but a better choice is to exploit an angular coordinate $s$ on the circle.

A material point of mass $m$ is constrained to stay on $\Gamma$ supposed to be frictionless. There is no reference frame where this constraint does not depend on time. We can use the parameter $s$ as Lagrangian coordinate. If there are no forces in addition to the reactive force due to the constraint, we can describe the equation of motion in terms of Euler-Lagrange equations of the Lagrangian $$L(t, s, \dot{s}) = T(t,s,\dot{s})\:.$$ where $$T = \frac{m}{2}{\bf v}(t,s, \dot{s})^2$$ and $${\bf v}(t,s, \dot{s}) = 2ct (\cos s {\bf e}_x + \sin s {\bf e}_y) + (r+ct^2)\dot{s} (-\sin s {\bf e}_x + \cos s {\bf e}_y)\:.$$ So that $$L(t, s, \dot{s}) = 4c^2t^4 +(r+ct^2)^2 \dot{s}^2 + (2 ct^2- (r+ct^2) \dot{s}) \sin (2s)\:.$$ The Euler-Lagrange equation for the curve $s=s(t)$ reads $$\frac{d}{dt}\left(2(r+ct^2)^2 \frac{ds}{dt} - (r+ct^2) \sin (2s) \right)- 2\left(2 ct^2- (r+ct^2) \frac{ds}{dt}\right) \cos (2s(t))=0\:.$$

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