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Newton's second law implies basically that the evolution of a mechanical system is completely determined as soon as the particles' initial positions $x(0)$ and velocities $\dot x(0)$ are specified. Since this is a second order equation, the corresponding Cauchy problem $$\begin{cases}\ddot x(t) = f(x(t),\dot x(t), t), &\ t>0 \\ x(0)=x_0, \\ \dot x(0)=y_0,\end{cases}\qquad\qquad(*)$$ has a unique local in time solution for any initial data $(x_0,y_0)$ if $f(x,y,t)$ is a Lipschitz function w.r.t. $(x,y)$ (which is almost always the case in applications).

As for possible complexity of motions, there are no obstacles to completely chaotic behaviour of solutions to system $(*)$ whatsoever, provided that $f(x,y,t)$ is a nonlinear function w.r.t $(x,y)$ and any of the following conditions is satisfied:

• the number of interacting particles $n\geq 3$;
• the forcing term is nonconservative;
• the system is essentially nonautonomous (i.e. $\partial_t f(x,y,t)\neq0$ identically).

Just google "chaos in classical mechanics" for numerous examples of innocent looking mechanical systems exhibiting wildly exotic dynamics.

Edit. See also a somewhat related MO question Does every ODE comes from something in physics?.

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Newton's second law implies basically that the evolution of a mechanical system is completely determined as soon as the particles' initial positions $x(0)$ and velocities $\dot x(0)$ are specified. Since this is a second order equation, the corresponding Cauchy problem $$\begin{cases}\ddot x(t) = f(x(t),\dot x(t), t), &\ t>0 \\ x(0)=x_0, \\ \dot x(0)=y_0,\end{cases}\qquad\qquad(*)$$ has a unique local in time solution for any initial data $(x_0,y_0)$ if $f(x,y,t)$ is a Lipschitz function w.r.t. $(x,y)$ (which is almost always the case in applications).

As for possible complexity of motions, there are no obstacles to completely chaotic behaviour of solutions to system $(*)$ whatsoever, provided that $f(x,y,t)$ is a nonlinear function w.r.t $(x,y)$ and any of the following conditions is satisfied:

• the number of interacting particles $n\geq 3$;
• the forcing term is nonconservative;
• the system is essentially nonautonomous (i.e. $\partial_t f(x,y,t)\neq0$ identically).

Just google "chaos in classical mechanics" for numerous examples of innocent looking mechanical systems exhibiting wildly exotic dynamics.