The purpose of mechanics is to describe how bodies change their position in space with ``time.'' I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations. Let us proceed to disclose these sins.
Albert Einstein, Relativity, the Special and General Theory , p. 9
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.
A pin thrown by a juggler takes a rather predictable path and rotates in a rather predictable way. In fact, the skill of juggling depends crucially on this predictability. It is also a remarkable discovery that the same mathematical tools used to describe the motions of the planets can be used to describe the motion of the juggling pin.
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.
There are many conceivable ways a system could move that never occur. We can imagine that the juggling pin might pause in midair or go fourteen times around the head of the juggler before being caught, but these motions do not happen. How can we distinguish motions of a system that can actually occur from other conceivable motions? Perhaps we can invent some mathematical function that allows us to distinguish realizable motions from among all conceivable motions.
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 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 the 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.
In the Newtonian formulation the forces can often be written as derivatives of the potential energy of the system. The motion of the system is determined by considering how the individual component particles respond to these forces. The Newtonian formulation of the equations of motion is intrinsically a particle-by-particle description.
In the variational formulation the equations of motion are formulated in terms of the difference of the kinetic energy and the potential energy. The potential energy is a number that is characteristic of the arrangement of the particles in the system; the kinetic energy is a number that is determined by the velocities of the particles in the system. Neither the potential energy nor the kinetic energy depends on how those positions and velocities are specified. The difference is characteristic of the system as a whole and does not depend on the details of how the system is specified. So we are free to choose ways of describing the system that are easy to work with; we are liberated from the particle-by-particle description inherent in the Newtonian formulation.
The variational formulation has numerous advantages over the Newtonian formulation. The equations of motion for those parameters that describe the state of the system are derived in the same way regardless of the choice of those parameters: the method of formulation does not depend on the choice of coordinate system. If there are positional constraints among the particles of a system the Newtonian formulation requires that we consider the forces maintaining these constraints, whereas in the variational formulation the constraints can be built into the coordinates. The variational formulation reveals the association of conservation laws with symmetries. The variational formulation provides a framework for placing any particular motion of a system in the context of all possible motions of the system. We pursue the variational formulation because of these advantages.
1 A stationary point of a function is a point where the function's value does not vary as the input is varied. Local maxima or minima are stationary points.
2 The variational formulation successfully describes all of the Newtonian mechanics of particles and rigid bodies. The variational formulation has also been usefully applied in the description of many other systems such as classical electrodynamics, the dynamics of inviscid fluids, and the design of mechanisms such as four-bar linkages. In addition, modern formulations of quantum mechanics and quantum field theory build on many of the same concepts. However, it appears that not all dynamical systems have a variational formulation. For example, there is no simple prescription to apply the variational apparatus to systems with dissipation, though in special cases variational methods can still be used.