Does the Cauchy-Lipschitz theorem extend to higher order DEs?

Question

The Cauchy-Lipschitz theorem says that the particular solution to a linear first order is unique provided $(x,y)$ are continuous over the domain. Does this extend to higher order DEs? Also does this apply to partial differential equations?

Not being a mathematician I have to think in terms of physical applications. For instance in classical mechanics would a unique solution to Hamilton's equations (linear first order PDEs) imply a unique solution to the Lagrange equation (linear, second order ODE)?

Answer 1

Cauchy Lipschitz theorem extend nicely on higher order ODE, by writting

$$Y(t)= (y(t),y'(t),\cdots, y^{(n-1)}(t))$$

You have then

$$\frac{\partial Y}{\partial t} (t) = F( Y(t), t) $$

And you can now apply Cauchy-Lipschitz (just be carefull, $F$ is now at value in $\mathbb{R}^{(n-1)}$ instead of $\mathbb{R}$, but the theorem is still valid)

There is no equivalent to Cauchy-Lipschitz theorem for the PDE : they are far more tricky and each kind of PDE has its own theorems.