Simple existence/uniqueness proof for Newton's 2nd?

[Fredrik]

Is there a simple proof of the existence and uniqueness of a solution of $x''(t)=f(x'(t),x(t),t)$, where f is an appropriately "nice" function? (Given an initial condition $x(t_0)=x_0,\ x'(t_0)=v_0$, or a boundary condition $x(t_0)=x_0,\ x(t_1)=x_1$).

 

[Office_Shredder]

It's called Picard's theorem. It relates to first order differential equations, but through using vectors can solve a system of them. Any second order differential equation can be turned into a system of first order differential equations by the following (using your differential equation as the starting point)

$$z_1(t)=x(t)$$ and $$z_2(t)=x'(t)$$ we get

$$z_1'(t)=z_2(t)$$

and

$$z_2'(t)=f(z_1(t), z_2(t),t)$$

Here's a wikipedia link to the theorem
http://en.wikipedia.org/wiki/Picard–Lindelöf_theorem

 

[Fredrik]

Thanks. I've seen references to Picard's theorem and I own a book that proves it, but I thought it was more general than that. I was hoping that there would exist a theorem with a particularly simple proof that covers the cases that are interesting for physics, but I guess not.

What theorem do we use when we've been given boundary conditions instead of an initial condition? There's still a unique solution, right?

 

[Office_Shredder]

When you're given boundary conditions it's a lot tougher. I forgot to mention a general technique for solving odes:
http://en.wikipedia.org/wiki/Method_of_variation_of_parameters

But you can have problems. Let's take the easy equation $$y''+y=0$$. So $$y=Acos(x)+Bsin(x)$$. Let's furthermore take the boundary conditions $$y(0)=0$$ and $$y(\pi)=0$$. Then all we get is that the solution is something of the form $$y=Asin(x)$$

If instead we had $$y(0)=0$$ and $$y(\pi)=1$$ there would be no solution.