Basic Equations of Point Mechanics

Mechanics is the oldest physical discipline. Its ideas, however, have influenced many other branches of physics. The goal in this chapter is to present some general principles of point mechanics which are necessary to understand elasticity theory, hydrodynamics, and many other branches of physics (statistical physics, theory of relativity, electrodynamics, quantum mechanics and quantum field theory, etc.). We will try to explain the close relation between the results about variational problems of Part III and the basic principles of mechanics. In particular, we explain the connection between Lagrange’s multiplier rule and the principle of least constraint and the least (stationary) action. To introduce the reader to the basic ideas, we consider in Section 58.2 a simple, but typical example: equilibrium state and motion of a balance, and its stability. Many modern expositions begin with the principle of stationary action. This principle, however, does not explicitly contain the most important physical concept—the force. Also, the principle of the stationary action, other than the principle of the least constraint, does not admit the most general side conditions, with nonlinear relations for the velocities.