Introduction

The laws of physics were established from experimental observations by finding simple relations to describe these observations; for example, the hypothesis of linearity between the flux and the force was established as a general principle of thermodynamics. Once a relation has been chosen, a coefficient of proportionality is selected. This coefficient is typically viewed as an intrinsic physical property, even though it depends directly on the choice of equations. If the law is overly simplistic, inconsistencies will be encountered when we attempt to generalize to parameter intervals larger than those originally considered. Fortunately, the complexity of the observed physical behavior typically allows us to reconstruct enough consistency to cover a broad spectrum of the parameter space. If only the thermodynamic coefficients themselves have any true physical significance, why should we need to introduce an ideal gas law between the variables, other than for convenience?

The laws of mechanics are some of the oldest laws in physics; throughout history, Galileo, Issac Newton and Albert Einstein each left a prodigious mark on our understanding of the universe around us. The concepts of force, mass and acceleration - linked together by Newton's second law - have not changed in modern mechanics. Over the past three centuries, a great succession of physicists, mathematicians and engineers gradually perfected the laws and equations of mechanics, which today seem inexorably set in stone. Leonhard Euler, Maurice Couette, Daniel Bernoulli, Ernst Mach, George Gabriel Stokes, Henri Navier, Clifford Truesdell and many others contributed to formalizing the laws of mechanics within a modern mathematical framework. Much has been written on the epistemology of the connections between the theories of these famous individuals, and occasionally the personal relationships between them. The path taken by scientific thought over time seems natural and logical if we consider the path taken by its equations, from classical mechanics to relativistic mechanics. Sometimes, it is not entirely clear where exactly new ideas came from, especially during periods where publication could not always be taken for granted and religion sometimes interfered with the propagation of new theories. Together, this scientific work led to equations of mechanics that have been frozen in time for decades or perhaps even centuries; today, these equations allow us to simulate all kinds of motion of solids, fluids, propagating waves, etc., extremely realistically. Yet, the equations of each type of motion are very different, even though continuum mechanics was supposed to establish a unified representation of all physical phenomena within the same set of equations.

How might we have established the laws of mechanics from scratch if we had access to every observation in history? This book attempts to give an answer. We start with the following question: why did Isaac Newton formulate his fundamental laws of dynamics as an equality of forces, given that he was aware of the equivalence between inertial mass and gravitational mass? This equivalence principle is one of the cornerstones of Albert Einstein's theory of general relativity. Discrete mechanics also relies heavily on this principle, using it to eliminate the concept of mass, which is not required to describe an equality between accelerations in mechanics. The notions of rest mass and relativistic mass at the heart of relativistic mechanics clearly characterize mass as equivalent to energy. Similarly, the fundamental law of dynamics can be formulated as an equality between the acceleration experienced by a particle and the sum of the accelerations applied to this particle.

The intrinsic nature of the acceleration is not the same as that of the velocity; we can apply an absolute acceleration even if the absolute velocity of a body is not known. The classical approach to finding the absolute velocity is to consider an inertial frame of reference. We must accept that it is pointless to attempt to understand this concept of absolute velocity, as well as the trains, station masters and elevators considered by the various thought experiments of the last century. It must be abandoned; we cannot detect uniform motion. By contrast, we can now measure acceleration to extremely high accuracy, corroborating ever further that equality between masses truly represents a fundamental principle.

Discrete mechanics also abandons the idea of a continuum; the differential calculus and differentials introduced by Gottfried Wilhelm Leibniz and Isaac Newton played an essential role in formulating the laws of mechanics into their modern differential form. To reduce every variable to a point, we are forced to construct an inertial frame of reference so that we can compute derivatives in specific directions of space. If we abandon the continuum, we must therefore also abandon classical differentiation and integration. We can still scale the discrete topology down geometrically to orders of magnitude that preserve compatibility with our macroscopic view of matter. The operations of differentiation and integration are replaced by operators based on discrete differential geometry. Inertial frames of reference are replaced by local frames, in which each point of the domain only perceives an immediate neighborhood of points, edges and faces with known distances and orientations. Every point is connected by causality links defined within space-time. Nature provides plenty of examples of collective behavior: schools of sardines, flocks of starlings, etc. The presence of limits or obstacles is perceived by each object through the behavior of its immediate neighborhood over a period of time that reflects the causality between them.

The distinction between the material velocity of the medium and the wave velocity (celerity) can be preserved by viewing the latter as a parameter determined by local conditions, whereas the material velocity itself is simply a secondary variable, defined as a velocity field known only up to uniform motion. The wave velocity is bounded by the speed of light in a vacuum by the axiom proposed by Albert Einstein, but is a function of the local conditions in general. The material velocity itself is not bounded a priori, since none of our axioms directly imply that any such bound exists. Even though the laws of discrete mechanics can be applied to problems derived from cosmology, they will chiefly be applied to areas of classical mechanics where cause-and-effect relations are associated with finite wave velocities.

The local equilibrium hypothesis is also set aside; none of our axioms imply that an arbitrary medium is in local equilibrium, and various counterexamples can be found. Hence, state equations become useless; applying these equations generates artifacts and violates conservation equations such as the law of conservation of mass. In continuum mechanics, the state equations are typically used to close the system of equations so that it has one equation for each variable. Today, this approach seems simplistic and reveals a lack of understanding of the degree of autonomy of the equations of motion. Only some of the physical properties are worth knowing at any cost; these properties should influence the solution of course, but only via the relation that exists between the variables in the conservation equations. Including other constitutive equations among the laws of mechanics, such as rheology describing a material's behavior, is no longer justified. Like any other properties, these equations should simply be known locally and instantaneously, even if they depend on the problem variables. Thus, a drastic distinction is drawn and preserved between properties and conservation equations.

Is the tensor formulation of the equations of mechanics strictly necessary? Tensors, introduced by Woldemar Voigt in their modern form and adopted by many other renowned thinkers over the last century, were in particular employed by Albert Einstein to formulate his theory of relativity, with help from Marcel Grossmann. In the past, physicists and engineers have legitimately needed a representation of the properties of certain materials that varies as a function of the direction. But can we justify viewing tensors as an integral part of the laws of mechanics? After all, Maxwell's equations can be expressed equivalently in either vector or tensor form. Tensors are required to define the stress as the product of the gradient of the velocity or the displacement and a coefficient. Cauchy's symmetrization of the velocity tensor made matters worse. The generalization of vectors to tensors can of course be justified mathematically. But in the equations of motion, the use of tensors generates artifacts that require us to impose additional constraints to resolve the resulting indeterminacy. For example, in solid mechanics, compatibility conditions must be imposed before the displacement can be computed from the stress. Over the past two centuries, other artifacts of similar type have been introduced into the equations of continuum mechanics. Discrete mechanics adopts a different perspective of the notion of stress. Shearing in a plane can, for example, be induced by a rotational stress in the direction normal to this plane; continuum mechanics would describe the same shearing as a stress in the plane itself. Eliminating tensors from the formulation of the equations of motion is a fundamental aspect of the discrete mechanical approach. Setting aside the concepts of frame of reference and the dimensionality of space allows us to introduce the operators of differential geometry ad hoc.

The inappropriate usage of certain laws of physics...