Yesterday, I posted an image of the #LorenzAttractor showing the evolution of three trajectories (shown in red, green and blue) starting close together. Here, I've made it into a little animation to show how the paths initially stay close to each other but after about a quarter of the duration plotted, they #diverge from each other irrevocably (i.e. become uncorrelated) but remain part of the #ChaoticAttractor.

#DynamicalSystems #ChaoticAttractors #StrangeAttractors #NumericalSolutions #Mathematics #AppliedMathematics #CCBYSA #FreeSoftware #WxMaxima

Just messing about a bit. Here is the famous #LorenzAttractor plotted using #WxMaxima. The three #trajectories, shown in red, green and blue are for three fairly nearby #InitialConditions.

#DynamicalSystems #ChaoticAttractors #StrangeAttractors #NumericalSolutions #Mathematics #AppliedMathematics #CCBYSA #FreeSoftware

Following on from yesterday's posting of flow around a #ZhukovskyAerofoil, I've added #equipotential lines (blue) in addition to #streamlines (black). The two families are #contours of the #real and #imaginary parts of an #AnalyticFunction and therefore have the property of intersecting #orthogonally. The plots aren't ideal as there is some anomalous features where the equipotential lines are very close to the #aerofoil but it's still not bad.

Here, each frame is a different #wing profile, obtained this time by keeping the imaginary eccentricity of the generating #circle constant and varying only the real part.

#MyWork #CCBYSA #AppliedMathematics #Aeronautics #Aerodynamics

The #Zhukovsky #Aerofoil (sometimes transliterated as #Joukowsky from #Russian), is a 2D model of #streamlined #Airflow past a #wing. It uses #ComplexVariable and is an #AnalyticFunction (i.e. #Differentiable everywhere, save at isolated #Singularities). Take a circle in the #ComplexPlane which is not quite centred at the #origin but passes through the #coordinate (1,0) or (z=1+0i). Using the mapping w -> z+1/z, you get something that looks remarkably like an #aerofoil.

It is a #ConformalMapping meaning that angles are preserved during the mapping. In this animation, I've varied the imaginary part of the the eccentricity, while keeping the real part the same. With a zero #AngleOfAttack, you can see the change in the airflow around the #wing as its shape changes.

#MyWork #CCBYSA #AppliedMathematics #WxMaxima #FreeSoftware #Aeronautics #Aerodynamics #LaminarFlow

Forgive the recent apparent obsession (I'd call it a fascination) with the #cycloid but I've just discovered something I'd not heard of before. It is also called a #TautochroneCurve or #Isochrone curve, which means that a particle starting from any location on the curve will get to the #MinimumPoint at precisely the same time as a particle starting at any other point.

Here's an #animation I wrote today in #Maxima which illustrates the property.

#Dynamics #Kinematics #Mathematics #AppliedMathematics #Mechanics #ClassicalMecanics #WxMaxima #FreeSoftware #MyWork #CCBYSA

A couple of weeks ago, I posted an #animation of a point on a circle generating a #cycloid.

If you turn the curve "upside down", you get the #BrachistochroneCurve. This curve provides the shortest travel time starting from one cusp to any other point on the curve for a ball rolling under uniform #gravity. It is always faster than the straight-line travel time. This is an interesting problem in #ClassicalMechanics and exercised luminaries like #Newton and #Euler. I think the latter's use of the #CalculusOfVariations is a stroke of genius.

Anyway, the #animation took a bit of thought as it requires a bit of #Mechanics, some #Integration and is made a bit more tricky as the curve is multi-valued and so you need to treat different branches separately. The #AnimatedGif was produce with #WxMaxima.

For some reason that I can't fathom, I'm not finding it possible to upload the graphic and so if you want to see it, please use the link below.
https://drive.google.com/file/d/1edcZnZ_uiaQOQrFB03XHslNUVCmxcSut/view?usp=drive_link

#MyWork #CCBYSA #Mathematics #Maths #AppliedMathematics #Physics #Calculus

Imagine a circular wheel rolling, without skidding, on a flat, horizontal surface. The #locus of any given point on its #circumference is called a #cycloid. It is a #periodic #curve over a length equivalent to the #circle's circumference and has #cusps whenever the point is in contact with the surface (i.e. the two sides of the curve are tangentially vertical at that point).

Interestingly, it is also the curve that solves the #Brachistochrone problem, which means that starting at a cusp on the inverted curve (maximum height), a frictionless ball will roll under uniform gravity in minimum time from the start to any other point on the curve, even beating the straight line path.

#Mathematics #Geometry #Maths #AppliedMathematics #Mechanics #Kinematics #Dynamics #Physics #MyWork #CCBYSA #WxMaxima

A few weeks ago, I posted an example of a #projectile #trajectory in a uniform #GravitationalField without #AirResistance. It's a school-level exercise but it was nice to produce an #animation for it. To model air resistance is a bit more tricky as any #fluid, including #air, is complex and does not necessarily behave as you might expect. A simple model for it might be to assume it is a #linear #viscous fluid offering resistance proportional to the velocity of the projectile, i.e. R = -k.v, where v is the velocity vector and k is a constant which dictates how viscous the fluid is.

Happily, this model has exact solutions so I didn't need to do any numerical integration to find the form of the displacement. Here are four different cases, including the "ideal" case k = 0. The others are k = 0.1, 1 and 10, with the middle one highlighted. The essential difference between the non-resistance case and the others is that horizontal velocity steadily decreases with time and decays towards zero exponentially. The vertical velocity is also reduced but still grows. The most viscous case (k = 10) shows terminal velocity behaviour very quickly and the projectile appears to be moving through treacle.

#MyWork #CCBYSA #AppliedMathematics #Maths #WxMaxima