Newtonian dynamics

Lore & Background

Newtonian dynamics is often narrowed to Newton's second law, expressed as m a = F. In a multidimensional space, for N particles with masses m1,...,mN in three-dimensional Euclidean space, their motion is governed by Newton's second law applied to each particle. The three-dimensional radius-vectors and velocity vectors can be built into a single n=3N-dimensional radius-vector and velocity vector, respectively.

Reader's Guide

Newtonian dynamics provides the foundation for classical mechanics, describing the motion of particles under forces. Its mathematical generalizations allow for application in multidimensional and curved spaces, extending beyond the typical flat three-dimensional Euclidean space. The concept of configuration space and phase space, both Euclidean, is central to the formulation, where the kinetic energy of a single multidimensional particle with unit mass equals the sum of kinetic energies of the three-dimensional particles. Constraints, such as holonomic and scleronomic ones, reduce the degrees of freedom, leading to a constrained configuration space defined as an n-dimensional manifold within the original configuration space. Internal coordinates are used in Lagrangian mechanics to express the radius-vector as a function of these coordinates, resolving constraint equations identically.

Did You Know?

• Newtonian dynamics can be generalized to multidimensional and curved spaces.
• The configuration space of a Newtonian dynamical system is a flat multidimensional Euclidean space.
• Constraints of the form scalar equations are called holonomic and scleronomic.
• Each holonomic and scleronomic constraint reduces by one the number of degrees of freedom.

The Core Concept: Viscosity That Defies a Single Number

In a Newtonian fluid, the relationship between shear stress and shear rate is perfectly linear, passing through the origin, with the slope representing a fixed coefficient of viscosity. Non-Newtonian fluids break this simple proportionality. Their resistance to flow shifts in response to the forces applied, meaning no single constant can capture their behavior. A jar of ketchup, for instance, resists the first pull of the spoon yet pours freely once shaken. The same principle governs custard, toothpaste, starch suspensions, paint, blood, melted butter, shampoo, and countless salt solutions or molten polymers. In some cases the viscosity even drifts over time under a steady load, making the notion of a fixed coefficient of viscosity entirely inapplicable. What unites all these substances is that the link between the stress you apply and the rate at which the material deforms is no longer a straight line through zero. The fluid may thicken, thin, or require a threshold force before it moves at all, depending on which category of non-Newtonian behavior it exhibits.

A Taxonomy of Unusual Flow

Non-Newtonian behavior splits into several distinct families. Shear-thickening, or dilatant, fluids grow more resistant as the shear rate climbs; a slow stir reveals a milky, moderately viscous liquid, while a vigorous agitation makes it feel almost solid. The mirror image, shear thinning or pseudoplastic flow, sees viscosity drop under faster deformation—wall paint flows off a brush easily yet does not drip, and blood thins as it accelerates through vessels. Bingham plastics occupy a middle ground: they demand a finite yield stress before any flow begins at all, so their stress-versus-strain plot never touches the origin. Clay suspensions, drilling mud, toothpaste, mayonnaise, chocolate, and mustard all belong here, and their still surfaces can hold sharp peaks rather than lying flat. On the time-dependent side, rheopectic fluids require steadily increasing stress to sustain a given strain rate, while thixotropic fluids thin out over time and need less stress to accomplish the same deformation.

Oobleck, Flubber, and the Kitchen Cabinet

The most celebrated classroom demonstration of non-Newtonian physics is oobleck, a simple suspension of corn or potato starch in water mixed at roughly one part water to one-and-a-half or two parts starch. The name borrows from Dr. Seuss's Bartholomew and the Oobleck. Because of its dilatant character, oobleck lets a person walk across a large tub without sinking, provided each step lands with enough force to trigger the thickening. Placed atop a powerful subwoofer, it swells into standing waves in response to low-frequency sound, and a sharp punch makes it behave momentarily like a solid before it relaxes back into a pourable liquid. Another household favorite is flubber, or slime, made from polyvinyl acetate school glue and borax; it flows under gentle pressure yet fractures under stronger stress. Beyond these demos, the same physics hides in soap solutions, cosmetics, butter, cheese, jam, yogurt, saliva, mucus, magma, cement slurry, and paper pulp.

Beyond a Single Number: The Mathematics of Complex Flow

Because a single viscosity coefficient cannot describe these materials, researchers turn to a richer set of rheological properties that relate the full stress and strain-rate tensors across many flow conditions, including oscillatory shear and extensional flow. These measurements are carried out with specialized instruments called rheometers, and the underlying mathematics takes the form of tensor-valued constitutive equations drawn from continuum mechanics. Three canonical time-dependent models are the Oldroyd-B fluid, Walters' Liquid B, and the Williamson fluid, each named for the authors who defined them. For a time-dependent Ladyzenskaya-type model featuring a non-linear, velocity-dependent stress tensor, analysts have been unable to extract closed-form solutions, though a rigorous existence theorem guarantees that solutions do in fact exist. By contrast, the time-independent non-Newtonian case enjoys a much broader catalogue of known analytic solutions, making it a more tractable starting point for both theory and engineering design.