# On the existence and uniqueness of solutions of Hamiltonian differential equations

Let $$(M,\omega)$$ be a symplectic manifold, and $$H : M \times [0,1] \to \mathbb{R}$$ be a smooth time-dependent Hamiltonian on $$M$$. Then non degeneracy of $$\omega$$ implies the existence of a time-dependent Hamiltonian vector field $$X_H^t$$, uniquely defined by $$\omega(X_H^t,.) = dH(.,t).$$ Consider the differential equation $$\dot{x}(t) = X_H^t(x(t)).$$ If I am not mistaken, for any given $$x \in M$$, there exists a unique solution with initial condition $$x(0) = x$$ defined on an interval containing $$0$$ (which depends on $$x$$), provided that $$H(.,t)$$ has Lipschitz derivatives. This way, we obtain a one-parameter family $$t \in I \to \phi_H^t(x) \in M$$, where $$I$$ is the maximal interval of existence, that is the intersection of all intervals containing $$0$$ on which the solutions are defined. The family $$\phi_H^t$$ is called the Hamiltonian flow generated by $$H$$.

My question regarding the above is the following:

• In the literature, one never talks about the Lipschitz condition on $$H(.,t)$$, even when $$M$$ is compact, and moreover the time-$$1$$ maps $$\phi_H^1$$ (the Hamiltonian diffeomorphisms) are always considered without specifying if they even exist.

Is there a particular reason why the Lipschitz condition would not be necessary in this situation, and why the time-$$1$$ maps would always be defined ?

Thanks!

• What do they mean by smooth in the literature? And, on a compact manifold any (sufficiently regular) ODE has solutions that can be extended to the whole time component of their domain. Commented Apr 29, 2019 at 11:56
• Smooth means infinitely differentiable. In the compact case I know that there are no problems, my questions regard the non-compact case. Commented Apr 29, 2019 at 12:02
• So, everything is OK.: $\mathrm{d}H$ is $C^{\infty}$, too, hence locally Lipschitz, so the Picard–Lindelöf theorem applies. Incidentally, $H$ being continuous and having the space derivatives up to second order continuous does suffice. Commented Apr 29, 2019 at 12:06
• Don’t we need $dH$ to be uniformly Lipschitz ? Why does locally suffices here ? Moreover, what can we say about the interval of existence of the flow ? Commented Apr 29, 2019 at 12:40
• Because the existence and uniqueness of a solution to an IVP is a local property. A general feature is that the right-endpoint of the interval of existence is lower semicontinuous. To get the existence for all time values the global Lipschitz property suffices. Commented Apr 29, 2019 at 18:02

For the non-compact case, there clearly needs to be conditions on $$H$$. If $$H(x,p) = x^\alpha - p^\alpha,$$ then on the $$x = p$$ line we have $$\dot{x} \propto x^{\alpha-1}$$ so for $$\alpha > 2$$, we have finite time blowup.
Also, see 10.4. https://drive.google.com/file/d/1USe9SS33q1L9k3S3xsgR-oy_9VWhEz5r/view; for Floer theory on $$T^*M$$, you consider a "Lagrangian" $$L$$ defined on the tangent bundle, and you restrict to Hamiltonians that are dual to $$L$$ that are "Tonelli Lagrangians", a condition that explicitly stipulates that the flow exists for all $$t$$. You also further stipulate that $$H$$ grows quadratically in the $$q$$ variables in some sense.
• Yes consider the flow on $M \times [0,1]$ for $(X_H, 1)$. Commented Apr 29, 2019 at 19:15