Dynamic duos: the building blocks of dimensional mechanics

Mechanics studies the relationships between space, time, and matter. These relationships can be expressed in terms of the dimensions of length , time , and mass . Each dimension broadens the scope of mechanics. Historically, mechanics emerged from geometry, which considers quantities like lengths or areas, with dimensions of the form . With the Renaissance, quantities combining space and time were considered, like speed, acceleration and later diffusivity, all of the form . Eventually, mechanics reached its full potential by including "mass-carrying" quantities such as mass, force, momentum, energy, action, power, viscosity, etc. These standard mechanical quantities have dimensions of the form where x and y are integers. In this contribution, we show that, thanks to this dimensional structure, these mass-carrying quantities can be readily arranged into a table such that x and y increase along the row and column, respectively. Ratios of quantities in the same rows provide characteristic lengths, while those in the same columns yield characteristic times, encompassing a great variety of physical phenomena from atomic to astronomical scales. Most generally, we show that selecting duos of mechanical quantities that are neither on the same row nor column of the table yields dynamics, where one mechanical quantity is understood as impelling motion, while the other impedes it. The force and the mass are the prototypes of impelling and impeding factors, but many other duos are possible. We present examples from the physical and biological realms, including planetary motion, sedimentation, explosions, fluid flows, turbulence, diffusion, cell mechanics, capillary and gravity waves, and spreading, pinching, and coalescence of drops and bubbles. This review provides a novel synthesis revealing the power of scaling or dimensional analysis, to understand processes governed by the interplay of two mechanical quantities. This elementary decomposition of space, time and motion into pairs of mechanical factors is the foundation of "dimensional mechanics", a method that this review wishes to promote and advance. Pairs are the fundamental building blocks, but they are only a starting point. Beyond this simple world of mechanical duos, we envision a richer universe that beckons with an interplay of three, four, or more quantities, yielding multiple characteristic lengths, times, and kinematics. This review is complemented by online video lectures, which initiate a discussion on the elaborate interplay of two or more mechanical quantities.