## What is Integration By Parts? A 5 Minute Introduction

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 $$\int u dv=uv – \int v du$$ where u and v are functions of one variable; x,…

## A Pure Hamiltonian Proof of the Maupertuis Principle

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…

## An Example of Servo-Constraints in Mechanics

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…

## Explaining the General Brachistochrone Problem

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…

## Learn Basic Kinematics in Classical Mechanics

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…

## Presenting a Rare Kinematic Formula

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…

## Elementary Construction of the Angular Velocity

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…