Existence of solution
Solving the physical system = Solving the Initial Value Problem (IVP)
- - Find the solution of \(u'(t)=\mathcal{F}(u,t)\), \(t\geq 0\)
- - With initial condition \(u(t=0)=u_0\)
Cauchy-Lipschitz (/Picard-Lindelof) theorem states that
- If \(\mathcal{F}\) is Lipschitz with respect to \(u\) and continuous with respect to \(t\)
- Then there exists a unique solution \(u(t)\).
The solution is called the integral curve of the IVP.
- - \(\mathcal{F}\) can be seen as a vector field
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- (Vector field in 6D for \(p(t)\in\mathbb{R}^3,v(t)\in\mathbb{R}^3\))
- - Solution is a path along this vector field passing by \(u_0\)
