# What is geometric intuition behind joint variational equation.

Given a differential equation
˙x=F(t,x)\dot{x}=F(t,x)
where possibly x∈Cnx\in \mathbb{C}^n one often considers the equation of variation
˙y=∂F∂x|xp(t)y\dot{y}=\frac{\partial F}{\partial x}\bigg|_{x_p(t)}y where y∈TCny\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 xx 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 xx. It is very useful to consider operator variational equation
˙Φ(t)=A(t)Φ(t)\dot{\Phi}(t)=A(t)\Phi(t) where A(t)=∂F∂x|xp(t)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
˙Ψ(t)=−A∗(t)Ψ(t)\dot\Psi(t)=-A^{*}(t)\Psi(t)
which is a true indicator of existence of integral curves for original equation. One proves simple lemma.

Φ(t)Ψ∗(t)=I\Phi(t)\Psi^{*}(t)=I

What is the intuition behind the adjoint variational equation?

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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.
– Evgeny
yesterday

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.
– Predrag Punosevac
yesterday

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?
– Evgeny
yesterday

Yes, I meant first integrals of the system.
– Predrag Punosevac
yesterday

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