Questions tagged [constrained-dynamics]

A constraint is a condition on the variables of a dynamical problem that the variables (or the physical solution for them) must satisfy. Normally, it amounts to restrictions of such variables to a lower-dimensional hypersurface embedded in the higher-dimensional full space of (unconstrained) variables.

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Write Equations of motion of a point particle that moves on the surface of a paraboloid [on hold]

I'm not a physics major student and I call myself a total noob in physics, however, I have this issue to solve. The problem is that I don't even know where to start from. One little thing to add: this ...
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Infinity problem in basic classical mechanics problem

My question arises from a classical mechanics problem from a Hong Kong physics training programme: This is not a homework question as I am not asking about how to solve the problems in the image. As ...
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Virtual Work and Arbitrary unit vectors

I was recently watching a lecture on MIT OCW where they were talking about Newton's Laws for a pulley system. The professor solved a problem pretty quickly using a method called virtual work. I tried ...
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Conservation of total energy for a system with holonomic constraints

Consider a system with generalized coordinates $u_1, u_2$ and $u_3$ such that $u_1$ and $u_2$ are dependent through the following holonomic constraint \begin{equation} G(u_1, u_2)=0. \end{equation} It ...
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Particle sliding on a sphere with friction

This is a generalization of the question Particle sliding on a sphere when we also have friction given by $F_f = \mu N$. See the following figure: Before doing anything, we can imagine what friction ...
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Arnold's holonomic constraints being limits of potential energy

The following quote comes from Arnold's "Mathematical methods in mechanics" book: "We consider potential energy $U_N = Nq_2^2 + U_0(q_1, q_2)$, depending on parameter $N$ (which we will tend to ...
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Non-holonomic constraints, degree of freedom and generalized coordinates

If a system has $N$ coordinates and $M$ number of holonomic constraints then number of degree of freedom $=N-M$ and generalized coordinates $=N-M$ too. But if there are $k$ non-holonomic constraints ...
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Forces in a barbell bench press and other similar movements - what's really happening?

This should probably be solvable using Lagrangian mechanics, but I haven't learned that yet and so I would appreciate an explanation of what happens without referring to it, if possible, and ideally ...
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Trouble with grasping D'Alembert's Principle intuitively

In Newtonian mechanics we all learned that a mass accelerates when it is under the influence of a nonzero net force. However, I learned of D'Alembert's Principle today and it seems to oppose what I ...
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Lagrange Multiplier with inequality conditions

How can Lagrange multiplier method be used when inequality conditions are given instead of equality conditions?
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Constrained Curve in 3 Dimensions [closed]

I have a particle in a 3D space that moves on a curve of the function $$r(x)=\begin{bmatrix}x \\ x\sin(x) \\ \exp(x^2)\end{bmatrix}$$ I know that there must be 1 degree of freedom left thus $S = 3N-P$...
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Sign of Lagrange multiplier

Hello I have a short question. Say I would consider a pendulum and define the Lagrangian as usual being \begin{align} L = \frac{1}{2} m(\dot{x}^2 + \dot{y}^2) - mgy \color{red}{-} \lambda (x^2 + y^2 ...
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Ambiguous Constraint equation

This is from the solution manual of some problem of Kleppner's book. I didn't understand how the constraint equation came about to be. First of all, I don't see how that equation is equal to the ...
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How to resolve velocity components?

In the arrangement shown in the figure, the ends P and Q of an inextensible string move downwards with uniform speed $u$, pulleys A and B are fixed. With what speed does the mass M move upwards? My ...
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Non-relativistic limit of Hamiltonian for a free particle in general relativity

The Hamiltonian for a particle moving in a gravitational field can be taken as $$\mathcal{H} = \frac12 \sum_{\mu,\nu=0}^3g^{\mu\nu}(x)p_\mu p_\nu\tag{1}$$ as long as the parametrization is affine. ...
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From Goldstein: A virtual (infinitesimal) displacement of a system refers to a change in the configuration of the system as the result of any arbitrary infinitesimal change of the coordinates $\... 1answer 52 views How to determine whether a set of coordinates are independent and sufficient to determine the system completely? In Analytical mechanics, when we formulate our principles, in general, it is assumed that we start with a cartesian coordinate system, and then find some generalised coordinates$q_j$s they are all ... 4answers 246 views Question about holonomic constraints Goldstein says that when a system of$N$particles is subject to$k$holonomic constraints, the positions$\mathbf{r}_1, \dots, \mathbf{r}_N$can be parameterized by$3N - k$independent coordinates$...
Consider a system of $N$ particles. There are $C$ holonomic time independent constraints,  \begin{aligned} f_1(\mathbf{r}_1,\dots,\mathbf{r}_N) & =0 \\ f_2(\mathbf{r}_1,\dots,\mathbf{r}_N) & ...