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  1. wrobel

    I Basic question about variational calculus

    yes, and that is exactly the case under consideration.
  2. wrobel

    Tensor Differentiation

    Just some remarks. 1) ##b_{ij}x_j## is not a tensor at least this expression does not keep its shape under changes of variables 2) the operation ##\partial/\partial x_i## takes tensors to not-tensors 3) if only linear changes are considered ##x_i=c_{ij}x'_j## then everything is ok
  3. wrobel

    I D'Alembert's principle vs Hamilton's principle

    yes, that is the point that's a key mistake. They are not equivalent. In the case of nonholonomic constraints the Hamilton principle implies vaconomic equations. I can only again refer you to the books cited above.
  4. wrobel

    I D'Alembert's principle vs Hamilton's principle

    There is a funny story as well. I think only Russians know it. Very long ago before the famous many valued Landau and Lifshitz textbook appeared, Landau and Pitaevsky wrote a textbook on classical mechanics. This book contained many errors and it was completely smashed by Fok in his article...
  5. wrobel

    I D'Alembert's principle vs Hamilton's principle

    the solution ##x_*(t)## to the variational problem described in #11 satisfies the equation $$\frac{d}{dt}\frac{\partial \mathcal L}{\partial \dot x^k}-\frac{\partial \mathcal L}{\partial x^k}=0,\quad \mathcal L=L+\lambda_k(t)a^k_s\dot x^s$$ this equation contains ##\dot\lambda## and it is not...
  6. wrobel

    I Basic question about variational calculus

    for a function of a single variable that is the same
  7. wrobel

    A Something on Baire categories

    There is an assertion that follows from very general theorem directly and I do not understand if this assertion trivial or it may be of some interest. The assertion is enclosed below please comment
  8. wrobel

    I D'Alembert's principle vs Hamilton's principle

    Landau and Lifshitz put in the basis of mechanics the Hamilton principle. They particularly say: let ##x_*(t)## be a critical point of the functional $$x(\cdot)\mapsto \int_{t_1}^{t_2}L(x(t),\dot x(t))dt$$ in a class of functions ##x(t)## that satisfy nonholonomic constraints: $$a_i^j(x)\dot...
  9. wrobel

    I D'Alembert's principle vs Hamilton's principle

    nope for details see A. M. Bloch, J. Baillieul, P. Crouch, J. Marsden: Nonholonomic Mechanicsand Control (Interdisciplinary Applied Mathematics). Springer, 2000 .
  10. wrobel

    I D'Alembert's principle vs Hamilton's principle

    by the way, I am sorry for self citing but here I tried to write a clear text with explanations about nonholonomic variational principle https://arxiv.org/abs/2104.03913 nothing new just an essence from textbooks
  11. wrobel

    I D'Alembert's principle vs Hamilton's principle

    they also claim that they deduce the equations of nonholonomic mechanics from the Hamilton principle :(
  12. wrobel

    Equilibrium of a stiff plate on inclined planes

    I have not seen the thread carefully perhaps somebody has already proposed to consider the potential energy's critical points
  13. wrobel

    A Noether's theorem for finite Hamiltonian systems

    hope this will not appear to be a severe offtop here https://www.physicsforums.com/threads/about-the-noether-theorem.996808/
  14. wrobel

    I How does inertia, a property of mass, arise?

    indeed! physics and metaphysics must be separated with a high and enduring wall
  15. wrobel

    I How does inertia, a property of mass, arise?

    and what does it practically imply?
  16. wrobel

    I Bicycle physics: Keeping balanced...

    $$\{(\delta\theta,\delta\psi,\delta\varphi,\delta x,\delta y)\}=span(a,b,c)$$ $$a=(1,0,0,0,0),\quad b=(0,0,1,0,0)$$ $$c=(0,1,0,-R\cos\varphi,-R\sin\varphi)$$
  17. wrobel

    I Bicycle physics: Keeping balanced...

    I would use D'Alambert-Lagrange $$[L]_\theta:=\frac{d}{dt}\frac{\partial L}{\partial \dot\theta}-\frac{\partial L}{\partial \theta},$$ $$[L]_\theta\delta\theta+[L]_\psi\delta\psi+[L]_\varphi\delta\varphi+[L]_x\delta x+[L]_y\delta y=0$$
  18. wrobel

    I Bicycle physics: Keeping balanced...

    is not the stability\instability dependent on the velocity?
  19. wrobel

    I Bicycle physics: Keeping balanced...

    I think it would be a good exercise to write down equations of coin's rolling without slipping on a horizontal table and study stability of its straight vertical motion in the linear approximation. to begin with:
  20. wrobel

    What is the tension of the rope?

    I have to admit that my solution is completely incorrect. I am sorry
  21. wrobel

    What is the tension of the rope?

    I do not believe that. In my equations the total energy must decrease You can check it by yourself all the formulas are brought
  22. wrobel

    What is the tension of the rope?

    just look at the picture carefully: y is a length of the loop it surely decreases. The velocity of the chain's end is ##\dot x=-2\dot y## asymptotically one has $$\dot y\sim -\frac{1}{\sqrt{c_1-c_2t}},\quad c_1,c_2>0$$ as ##t\to c_1/c_2-##
  23. wrobel

    What is the tension of the rope?

    Ok then :) let's develop the variable mass version of the story. Equation from #11 takes the form $$-2\frac{d}{dt}(\dot y y)=gy.$$ After a change ##y^2=z## we have $$\ddot z=-g\sqrt z,$$ and let ##\dot z=u(z)## then $$\frac{1}{2}\frac{d}{dz}u^2=-g\sqrt z.$$ Eventually it follows that $$\dot...
  24. wrobel

    A The equations of variable mass systems

    now we can turn back to https://www.physicsforums.com/threads/what-is-the-tension-of-the-rope.1004390/#post-6506923 :)
  25. wrobel

    A The equations of variable mass systems

    I think you just missed ##\rho## and the formulas must be as follows
  26. wrobel

    A The equations of variable mass systems

    The equations of variable mass systems are usually deduced from some very informal argument. It is so at least for the books I know. So I tried to construct a formal proof based on the continuous media equations. Criticism, remarks etc are welcomed. Let ##D\subset \mathbb{R}^q## be an open...
  27. wrobel

    What is the tension of the rope?

    for the velocity the variable mass equation gives the same , I did not check for the force
  28. wrobel

    What is the tension of the rope?

    sure it is just incorrect to apply the standard Newton law to a variable mass system it is the acceleration of the center of mass who can not exceed g
  29. wrobel

    What is the tension of the rope?

    The one-dimensional fall of a folded chain with one end suspended from a rigid support and achain falling from a resting heap on a table is studied. Because their Lagrangians contain no explicittime dependence, the falling chains are conservative systems. Funny argument little bit. Why do that...
  30. wrobel

    What is the tension of the rope?

    Something like that I guess
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