Recent content by wrobel

  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?
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