#wxmaxima — Public Fediverse posts

Of course this was known to many famous #mathematicians, like #Euler and #Jacobi and they solved the equations in terms of #EllipticIntegrals. I find it satisfying to be able to visualize the #dynamics of the this system, using #FreeSoftware, such as #WxMaxima.

We’re having a #Soliton sale over here! Get yours quick! Seriously, though, this is a quick #animation of the #SolitonInteraction surface – the solution of the #Korteweg-deVries #Eauation showing (x,t,u(x,t)) giving you an all-round view.

#MyWork #CCBYSA #AppliedMathematics #Mathematic #Phyiscs #KdV #KdVEquation #FreeSoftware #WxMaxima

I posted on #Soliton interactions in the #Korteweg-deVries #Equation yesterday with an #Animation of how two #SolitaryWaves interact. The #PhaseShift experienced by both can also be seen in a static #3D plot with time, t, plotted on one axis. You should be able to see a “dog-leg” in the trajectory of each wave after it interacts with the other.

#MyWork #CCBYSA #Mathematics #AppliedMathematics #Physics #FreeSoftware #WxMaxima

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

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).

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

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).

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

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).

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

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).

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

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).

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

This is why #Engineers use #TransitionCurves in the form of #CornuSpirals or #Clothoids to connect from the straight segment onto a circular arc. It has the property that it starts with zero #curvature at the beginning and then the curvature increases linearly with distance travelled along the curve. At the appropriate curvature, it is connected to a circular arc with the same curvature.

#MyWork #WxMaxima #CivilEngineering #HighwayEngineering #CCBYSA

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.

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

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 with #period over the #circle's circumference and has #cusps whenever the point is in contact with the surface (the two sides of the curve are tangentially vertical at that point).

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

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. Here are four different cases, including the ideal case k = 0. The others are k = 0.1, 1 and 10, with the k=1highlighted. Pt2...

#MyWork #CCBYSA #Maths #WxMaxima

Here are #animated #gifs of the representation of #NumericalIntergration #approximations with increasing numbers of strips. Represented are the Rectangle Method, Trapezium Rule and Simpson’s Rule. Note how much faster the second and third methods improve the approximation the exact blue curve in each case as the number of strips increases. Produced using #wxmaxima. They are all available on #Wikipedia.

#MyWork #Mathematics #Maths #Numerics #RectangleMethod #TrapeziumRule #SimpsonsRule #CCBYSA