Recent content by wrobel

not always but in the absence of friction

Then it is indeed a simple application of the torque equation.

Is ##\ell## a rod that connected to the sphere perpendicular and such that they both form a rigid body?

You need the torque equation with respect to the point O to the whole system and a lot of geometry.

When you consider level sets ##\{(x,y)\mid f(x,y)=const\}## it is important to find critical points of the function ##f## and understand which kind these critical points are. So first find the points such that ##df=0##. It is like drawing a phase portrait of a Hamiltonian system with the...

I think you need a lemma from p.289 S Lang

The open set ##\Omega\subset\mathbb{R}^m## with the standard coordinates is a coordinate patch in this graph, so the graph is a manifold

There are articles by Alexandre Karapetyan where he shows that the nonholonomic constraints can be considered as a limit case of some field of viscous friction forces. Alexandre Karapetyan Realization of nonholonomic constraints by viscouse friction and stability of rotation of Celtic stone"...

Exactly! You do not have a functional to differentiate. Speaking informally, what you write is an infinite dimensional differential form of ##\delta q## and this differential form can not be presented as a differential of the Action functional for the nonholonomic case. Nobody doubts the...

Let's drop the dependence on ##t## in the function ##L## and in the constraints. Can you deduce your formulas from such a definition: $$\delta S:=\frac{d}{d\varepsilon}\Big|_{\varepsilon=0}\int L(q(t,\varepsilon), q_t(t,\varepsilon))dt=0,\quad a_{ij}(q(t,\varepsilon))q_t^i(t,\varepsilon)=0,$$...

Detail it please. It is the core of the whole of our issues. I gave the definition in #9 now it is important that you explain what you mean.

May I ask a question first. vanhees71: Please give a definition of a stationary point of the functional ##S## in the nonholonomic case.

I am sure that d'Alembert completely agrees with the experiment. That is not the point.

We have different views on this theory, so I prefer not to start the discussion again, but merely provide the topic starter with references and proofs. Let him decide by himself.