PhD - Interested in differential equations and classical mechanics
Definition/Summary In this article, we shall learn a method for integrating the product of two functions. This method is derived from the ‘product rule’ for differentiation, but can only be applied to integrate products of certain types. Equations [tex]\int u dv=uv – \int v du[/tex] where u and v are functions of one variable; x,…
Here is another version of proof of Maupertuis’s principle. This version is pure Hamiltonian and independent of the Lagrangian approach. The proof is based upon the Hamiltonian version of the Vector Field Straightening Theorem. It seems that such a style of exposition simplifies the understanding of this non-trivial construction. First, recall and briefly discuss…
Servo-constraint was invented by Henri Beghin in his Ph.D. thesis in 1922. For details see the celebrated monograph in rational mechanics by Paul Appell. To understand what this is, we consider the following example. A trolley of mass ##M## can move freely along the horizontal ground in the standard gravity field. A pendulum is placed…
Consider a problem about the curve of fastest descent in the following generalized statement. Suppose that we have a Lagrangian system $$L(x,\dot x)=\frac{1}{2}g_{ij}(x)\dot x^i\dot x^j-V(x),\quad x=(x^1,\ldots, x^m)\in M.\qquad (*)$$ Here ##M## is a smooth configuration manifold, ##\mathrm{dim}\,M=m##. The functions ##g_{ij}## form a Riemann metric in ##M##. Assume also that our system is constrained with the…
There is an interesting thing in teaching of Classical Mechanics. Several theorems which presented below form a core part of kinematics for all Russian textbooks in Classical Mechanics (excluding Classical Mechanics for physicists) but European and USA textbooks contain this material just partly and from time to time. Perhaps I just do not know enough…
Here we present some useful kinematic fact which is uncommon for textbooks in mechanics. Consider a convex rigid body (RB) rolling without slipping on a fixed plane. (The plane can actually be replaced with a some other fixed surface.) In the picture RB is a filled with dots oval . At a given moment of time…
Physics books seldom contain an accurate definitions of the angular velocity of a rigid body. I believe that the following construction is as simple as possible and also rigorous. Assume we have a rigid body which is moving in the space Theorem. In each moment there exists a unique (pseudo)vector ##\boldsymbol\omega ## such that for…