Book of Interest: Structure and Interpretation of Classical Mechanics

Structure and Interpretation of Classical Mechanics


From Chapter 1:

The subject of this book is motion and the mathematical tools used to describe it.

Centuries of careful observations of the motions of the planets revealed regularities in those motions, allowing accurate predictions of phenomena such as eclipses and conjunctions. The effort to formulate these regularities and ultimately to understand them led to the development of mathematics and to the discovery that mathematics could be effectively used to describe aspects of the physical world. That mathematics can be used to describe natural phenomena is a remarkable fact.

Classical mechanics describes the motion of a system of particles, subject to forces describing their interactions. Complex physical objects, such as juggling pins, can be modeled as myriad particles with fixed spatial relationships maintained by stiff forces of interaction.

The motion of a system can be described by giving the position of every piece of the system at each moment. Such a description of the motion of the system is called a configuration path; the configuration path specifies the configuration as a function of time. The juggling pin rotates as it flies through the air; the configuration of the juggling pin is specified by giving the position and orientation of the pin. The motion of the juggling pin is specified by giving the position and orientation of the pin as a function of time.

The path-distinguishing function that we seek takes a configuration path as an input and produces some output. We want this function to have some characteristic behavior when its input is a realizable path. For example, the output could be a number, and we could try to arrange that this number be zero only on realizable paths. Newton’s equations of motion are of this form; at each moment Newton’s differential equations must be satisfied.

However, there is an alternate strategy that provides more insight and power: we could look for a path-distinguishing function that has a minimum on the realizable paths—on nearby unrealizable paths the value of the function is higher than it is on the realizable path. This is the variational strategy: for each physical system we invent a path-distinguishing function that distinguishes realizable motions of the system by having a stationary point for each realizable path.1 For a great variety of systems realizable motions of the system can be formulated in terms of a variational principle.2

Mechanics, as invented by Newton and others of his era, describes the motion of a system in terms of the positions, velocities, and accelerations of each of the particles in the system. In contrast to the Newtonian formulation of mechanics, the variational formulation of mechanics describes the motion of a system in terms of aggregate quantities that are associated with the motion of the system as a whole.