section_path: Chaos and unpredictability/Nonlinear dynamics
section_name: Nonlinear dynamics
paragraph_text: Nonlinear dynamics
  
  Newton's laws of motion allow the possibility of chaos. That is, qualitatively speaking, physical systems obeying Newton's laws can exhibit sensitive dependence upon their initial conditions: a slight change of the position or velocity of one part of a system can lead to the whole system behaving in a radically different way within a short time. Noteworthy examples include the three-body problem, the double pendulum, dynamical billiards, and the Fermi–Pasta–Ulam–Tsingou problem.
  
  Newton's laws can be applied to fluids by considering a fluid as composed of infinitesimal pieces, each exerting forces upon neighboring pieces. The Euler momentum equation is an expression of Newton's second law adapted to fluid dynamics. A fluid is described by a velocity field, i.e., a function v (x, t) {\displaystyle \mathbf {v} (\mathbf {x},t)} that assigns a velocity vector to each point in space and time. A small object being carried along by the fluid flow can change velocity for two reasons: first, because the velocity field at its position is changing over time, and second, because it moves to a new location where the velocity field has a different value. Consequently, when Newton's second law is applied to an infinitesimal portion of fluid, the acceleration a {\displaystyle \mathbf {a} } has two terms, a combination known as a total or material derivative. The mass of an infinitesimal portion depends upon the fluid density, and there is a net force upon it if the fluid pressure varies from one side of it to another. Accordingly, a = F / m {\displaystyle \mathbf {a} =\mathbf {F} /m} becomes ∂ v ∂ t + (∇ ⋅ v) v = − 1 ρ ∇ P + f, {\displaystyle {\frac {\partial v}{\partial t}}+(\mathbf {\nabla } \cdot \mathbf {v})\mathbf {v} =-{\frac {1}{\rho }}\mathbf {\nabla } P+\mathbf {f},} where ρ {\displaystyle \rho } is the density, P {\displaystyle P} is the pressure, and f {\displaystyle \mathbf {f} } stands for an external influence like a gravitational pull. Incorporating the effect of viscosity turns the Euler equation into a Navier–Stokes equation: ∂ v ∂ t + (∇ ⋅ v) v = − 1 ρ ∇ P + ν ∇ 2 v + f, {\displaystyle {\frac {\partial v}{\partial t}}+(\mathbf {\nabla } \cdot \mathbf {v})\mathbf {v} =-{\frac {1}{\rho }}\mathbf {\nabla } P+\nu \nabla ^{2}\mathbf {v} +\mathbf {f},} where ν {\displaystyle \nu } is the kinematic viscosity.