Have there been any breakthroughs in finding exact solutions to physics equations?

[Locut0s]

Many if not most modern branches of physics are founded on equations that describe the systems they model extremely well. However in all but the most trivial of examples there is usually no known exact solution. For their application this is usually not an issue as perturbative methods usually exist that allow one to find solutions that are arbitrarily close to what they need given enough time and computing power. But I was wondering are there any really well known examples of exact solutions to modern physics equations in a more general case?

 

[CycloWizard]

Pretty much anything that can be solved analytically has been at this point, and most of those solutions have existed for decades. There are results published of particular solutions of various field equations to look at specific systems under specific conditions, but I'm not sure if that's what you're asking.

 

[Locut0s]

Originally posted by: CycloWizard
Pretty much anything that can be solved analytically has been at this point, and most of those solutions have existed for decades. There are results published of particular solutions of various field equations to look at specific systems under specific conditions, but I'm not sure if that's what you're asking.

That is what I meant. Things like the n body problem for even small n like n=3 or 4.

 

[DrPizza]

I'm not quite sure what you're asking either. As far as break throughs for specific solutions, there are already proofs that some answers cannot be expressed/found using the basic elementary functions. Search for the quintic for more information on one particular function that's been well written about. The only way to solve such equations is through numerical methods - methods for approximating solutions, not finding exact solutions. As far as a "breakthrough" - if there's proof that there's no possible way, then I don't think we'll see a "breakthrough."

 

[silverpig]

All but the most trivial ideal systems require at least some perturbation theory. Even the hydrogen atom has only been solved analytically if you ignore a lot of effects. You need perturbation theory to really get the right answers out of even hydrogen.

 

[CP5670]

It depends on what you call an exact solution, for that matter. At some level, you can always just introduce new special functions and define them to be the solutions.

 

[f95toli]

Originally posted by: Locut0s

Originally posted by: CycloWizard
Pretty much anything that can be solved analytically has been at this point, and most of those solutions have existed for decades. There are results published of particular solutions of various field equations to look at specific systems under specific conditions, but I'm not sure if that's what you're asking.

That is what I meant. Things like the n body problem for even small n like n=3 or 4.

Someone (I don't remember the name) came up with an analytical solution to the 3-body problem about 10 years ago. However, the solution is in the form of a very complicated series so it is useless for practical calculations.

Also, remember that being able to solve an equation analytically does not neccesarily give you more useful information than a numerical solution.
Analytical solutions to most PDEs usually involve special functions (spherical harmonics etc) which in themselves are rather complicated and their values can -in general- not be calculated without numerical methods. In the "old days" people just used math tables to find the values of those functions.
That said, analytical solutions wil often give you some idea about the "structure" of the solution, interesting limits etc.

 

[cirthix]

Originally posted by: CP5670
It depends on what you call an exact solution, for that matter. At some level, you can always just introduce new special functions and define them to be the solutions.

Then just make a numerical approximation to the new function :evil:

 

[Born2bwire]

Originally posted by: CP5670
It depends on what you call an exact solution, for that matter. At some level, you can always just introduce new special functions and define them to be the solutions.

Exact solution is the machine precision that I compile at. My computer always tells me the exact answer. Just today I calculated pi to 16 decimal points: 3.123456789047579.

 

[KillerCharlie]

There are only exact analytical solutions to the very simplest of fluid flows.

 

[CycloWizard]

Originally posted by: KillerCharlie
There are only exact analytical solutions to the very simplest of fluid flows.

Even better: there aren't even numerical solutions for many fluid flows. At least, not on useful time/length scales.

 

[firewolfsm]

This is one aspect of physics I never really understood but have seen mentioned on this forum often. Even with advanced simulations or programs, it's not possible to account for all forces and calculate any outcome for fluid flow?

 

[CycloWizard]

Originally posted by: firewolfsm
This is one aspect of physics I never really understood but have seen mentioned on this forum often. Even with advanced simulations or programs, it's not possible to account for all forces and calculate any outcome for fluid flow?

It is possible, but only for very, very short time scales. Under laminar flow conditions (i.e. low flow velocities), exact solutions are often possible. However, under turbulent conditions, accurate modeling is still rarely possible. The only way currently available to very accurately model turbulent flow is using molecular simulation, which requires picosecond steps in the time domain. There are approximations that work under special conditions that allow some degree of accuracy at the continuum level using computational fluid dynamics, but turbulence never really reaches steady state, is highly chaotic, and is intrinsically nonlinear in the inertial terms of the Navier-Stokes equations.

 

[Fox5]

Originally posted by: Born2bwire

Originally posted by: CP5670
It depends on what you call an exact solution, for that matter. At some level, you can always just introduce new special functions and define them to be the solutions.

Exact solution is the machine precision that I compile at. My computer always tells me the exact answer. Just today I calculated pi to 16 decimal points: 3.123456789047579.

i c whut u did thar