# What is geometric intuition behind joint variational equation.

Given a differential equation $$\dot{x}=F(t,x)$$ where possibly $x\in \mathbb{C}^n$ one often considers the equation of variation $$\dot{y}=\frac{\partial F}{\partial x}\bigg|_{x_p(t)}y$$ where $y\in T\mathbb{C}^n$ which is a linear non-autonomous equation of the first order. This is typically firstly done when the theorem about smooth dependence of the solution with respect to initial data is proven in elementary ODE class or theorems about differentiability of the flow or the solution with respect to $x$ are proven. It is also used to show that evolutionary flow of the equations in complex domain is analytic or possibly even conformal with respect to $x$. It is very useful to consider operator variational equation $$\dot{\Phi}(t)=A(t)\Phi(t)$$ where $A(t)=\frac{\partial F}{\partial x}\bigg|_{x_p(t)}$. The geometric intuition behind all this is very simple and one can read about it for example in Hirsch-Smale ODE book from 1972. It is lesser known outside control-theory and intelligibility communities that it is very useful to consider adjoint variational equation $$\dot\Psi(t)=-A^{*}(t)\Psi(t)$$ which is a true indicator of existence of integral curves for original equation. One proves simple lemma.

$$\Phi(t)\Psi^{*}(t)=I$$

What is the intuition behind the adjoint variational equation?

• Frankly speaking I don't understand what do you mean by "true indicator of existence of integral curves for original equation". It would be great if you could write more here. Oct 22, 2016 at 13:02
• Are you familiar with Alain Goriely book "Integrability and Non-integrability of dynamical systems"? He has a couple pages on the relation between the integrals of Adjoint equation and original equation. Oct 22, 2016 at 14:53
• No, I haven't heard about that book, thanks for reference. Still I think that you've meant not integral curves (because if rhs is well-behaved they always exist), but first integrals of system then? Oct 22, 2016 at 15:10
• Yes, I meant first integrals of the system. Oct 23, 2016 at 3:03