Is this curve well known?

Question

I consider the curve $c(t)=(x(t),y(t))$ in $\mathbb{R}^2$ such that

$\frac{d^2x(t)}{dt^2}=-(a\sin t+b)\frac{dy(t)}{dt}$

$\frac{d^2y(t)}{dt^2}=(a\sin t+b)\frac{dx(t)}{dt}$

$a,b\in\mathbb{R}$

Is the orbit curve of solution of above equations known?

Answer 1

It is not clear what you mean by "known" but this system can be solved explicitly, in quadratures of elementary functions. Set $x'=u,\; y'=v,\; g(t)=a\sin t+b$. Then your system becomes $$u'=-gv,\quad v'=gu.$$ Multiplying the first equation on $u$ and second on $v$ and adding, we obtain $u'u+v'v=0,$ therefore $u^2+v^2=c$. Then the first equation becomes $(u')^2=g^2(c-u^2)$, and this is a separable equation, with an explicit integral.