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There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
Important formulas in kinematics define the velocity and acceleration of points in a moving body as they trace trajectories in three-dimensional space. This is particularly important for the center of mass of a body, which is used to derive equations of motion using either Newton's second law or Lagrange's equations .
In fluid mechanics, a two-dimensional flow is a form of fluid flow where the flow velocity at every point is parallel to a fixed plane. The velocity at any point on a ...
The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781, [1] is defined for incompressible (divergence-free), two-dimensional flows. The Stokes stream function , named after George Gabriel Stokes , [ 2 ] is defined for incompressible, three-dimensional flows with axisymmetry .
For two-dimensional, plane strain problems the strain-displacement relations are = ; = [+] ; = Repeated differentiation of these relations, in order to remove the displacements and , gives us the two-dimensional compatibility condition for strains
The subproblem yields two solutions in the event that the circles intersect at two points; one solution if the circles are tangential; and no solution if the circles fail to intersect. Let ξ 1 {\textstyle \xi _{1}} and ξ 2 {\textstyle \xi _{2}} be two zero-pitch twists with unit magnitude and intersecting axes.
Smale's problems is a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 [1] and republished in 1999. [2] Smale composed this list in reply to a request from Vladimir Arnold, then vice-president of the International Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century.
The equation of motion for Stokes flow can be obtained by linearizing the steady state Navier–Stokes equations.The inertial forces are assumed to be negligible in comparison to the viscous forces, and eliminating the inertial terms of the momentum balance in the Navier–Stokes equations reduces it to the momentum balance in the Stokes equations: [1]