Time Dependent Semi-Major & Semi-Minor Axes in Ellipse Equation

[WannabeNewton]

Can the semi - major and semi - minor axes of an ellipse be time dependent? More specifically, can you have time dependent semi - major and semi - minor axes present in the standard form of the ellipse? I have an equation of the form [tex]\frac{(\xi ^{1}(t))^{2} }{a^{2}} + \frac{(\xi ^{2}(t))^{2}}{b^{2}} = 1 [/tex] where [itex]\xi ^{\alpha }[/itex] are components of a separation vector, [itex]a^{2} = [2 + \frac{1}{2}sin^{2}\omega t](\xi ^{1}(0))^{2}[/itex], and [itex]b^{2} = [2 + \frac{1}{2}sin^{2}\omega t](\xi ^{2}(0))^{2}[/itex] but I don't know if the standard form can actually have time dependent semi - major and minor axes.

 

[HallsofIvy]

Yes, of course. However what you are writing does NOT.
[tex]\frac{\xi^1(t))^2}{a^2}+ \frac{\xi^2(t))^2}{b^2}= 1[/tex]
is a single ellipse with axes of length a and b for all t. If t is "time", this could be interpreted as an object moving around that fixed ellipse with changing speed.

An ellipse with "time dependent sem- major and minor axes" would be
[tex]\frac{x^2}{(a(t))^2}+ \frac{y^2}{(b(t))^2}= 1[/tex]

You can add "time dependence" or dependence on any other variable at will, just by making some parameters function of that variable.

 

[WannabeNewton]

An ellipse with "time dependent sem- major and minor axes" would be
[tex]\frac{x^2}{(a(t))^2}+ \frac{y^2}{(b(t))^2}= 1[/tex]

So if in the original equation, If I had the usual X and Y instead of the time dependent separation vector but with the same form of the semi - major and minor axes that I posted above then it would be allowed? Thanks for the reply.

EDIT: Never mind I get what you are saying. I think I should have had the initial coordinate separations on the top and the time variance of the separation plus the sin terms on the bottom.