# Does existence of an analytic solution to an equation of motion given by Newton's second law depend on coordinates?

Newton's second law is a coordinate agnostic statement, we can use it to calculate the forces in a coordinate system, and hence, the motion of the body in that coordinate system. However, depending on the choice of the coordinate system, the difficulty maybe more or less. Sometimes the DE may itself look unsolvable in a certain coordinate system. This led me to wonder.. Are certain EOMS having analytic solution only in certain coordinate system?

• If you have a solution in some coordinate system you could presumably define a new set of coordinates using a Jacobian that does not have a closed form. I guess this would mean the solution in the new coordinates wouldn't have a closed form either. May 14, 2022 at 8:21

I don't think you're using "analytic" in the strict mathematical sense. You're probably thinking of "closed-form solutions in terms of elementary functions". Anyway, existence of solutions is a coordinate-independent thing.

On the other hand, let's say you solve the system of ODEs in one coordinate system, but you now decide to transform to the new coordinate system. If this change of coordinates (or its inverse depending on how you want to phrase it) can't be written in terms of elementary functions, then the transformed solutions won't be expressible in terms of elementary functions. But I repeat again: the existence/non-existence of solutions is a coordinate-independent assertion (but usually you're dealing with smooth ODEs so solutions exist and are unique once you provide the appropriate initial conditions).

• A solution can exist without being expressible in elementary functions..? O_O
– Babu
May 14, 2022 at 8:27
• @Aplateofmomos yes of course. this is definitely not a surprising thing. Imagine the ODE $f'(x)=e^{-x^2}$ (the simplest type of ODE. first order with no other complicated dependencies). The solution is obtained by the FTC: $f(x)=C+\int_0^xe^{-t^2}\,dt$, i.e this is essentially the error function, which is known to not be expressible in terms of elementary functions. ... this is so simple that even first year calculus students can write down the solution and claim it exists, but it just can't be written in terms of elementary functions May 14, 2022 at 8:30
• btw, we have very general theorems about existence and uniqueness of solutions to ODEs (search on Wikipedia for precise statements). May 14, 2022 at 8:31
• What would be an example of solution existing in terms of elementary functions but jacobian to transfer into another coordinate system being non elementary?
– Babu
May 14, 2022 at 8:33
• @Aplateofmomos derivatives of elementary functions (trig, powers, logarithms, and their inverses etc) are elementary, so that's not possible May 14, 2022 at 8:34

I'd say no, since once you have the solution in one coordinate system, you can always use coordinate transformation equations to then find the solution in the other coordinate system. For example, if I have the solution to a 2D problem in Cartesian coordinates, i.e., I know the functions $$x(t)$$ and $$y(t)$$, then I can easily find the solution in polar coordinates by substituting the known functions for the cartesian components in the transformation equations $$r(t)=\sqrt{x(t)^2+y(t)^2}$$, and $$\theta(t)=\arctan \left( {\frac{y(t)}{x(t)}}\right)$$.

Conversely, if an equation does not have an analytical solution in some coordinate system, it is obvious that there's no transformation you can do in order to make it have an analytical solution. If an equation of motion has a solution or not in a coordinate system, it will (or will not) have it in all coordinate system.

No! If there is a "analytic" solution for some equation of motion, at all, it may be displayed in any coordinate system, specially in a time-dependent one, with orientation of acting forces.