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"The Equation Breaks Down"

I remember hearing this statement in a physics documentary some time ago. I presume it to mean that an equation which gives valid or verifiable predictions within a certain range, once that range is exceeded, produces incorrect statements or values (for example Newtonian Physics cannot describe activity on a quantum level, therefore its mathematical predictions would "break down"). That an equation should "break down" on one level but remain true on another would seem to mean that the equation is not in fact true or absolute. I would like to know any other current mathematical examples in Physics where equations appear to collapse or produce seemingly nonsensical predictions. Surely the root of the collapse is in the model which cannot make sense of the equation's data, rather than the equation itself. Is this best solved empirically?

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Well, there's a difference between an equation 'breaking' and giving just a wrong answer. I wouldn't say giving a wrong answer that is theoretically possible is an equation "breaking down", but when you get literal nonsense like infinite probabilities, that's when I would say an equation "breaks down".

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Well, there's a difference between an equation 'breaking' and giving just a wrong answer. I wouldn't say giving a wrong answer that is theoretically possible is an equation "breaking down", but when you get literal nonsense like infinite probabilities, that's when I would say an equation "breaks down".

Could you give an example of an equation "giving a wrong answer that is theoretically possible". In that case the equation would have remained true or accurate within its own theoretical model, and hence not broken down, but nonetheless be "wrong" for predictions not valid in reality. In that case the equation most definitely does not break down. Still, it makes for unsatisfying predictions. I will have to look up infinite probabilities. If that is "literal nonsense" but still within an otherwise valid equation could it not just be an issue with our model of understanding which cannot make sense of it?

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Like a Galilean velocity addition at very high speeds. The answers are wrong, but give meaningful answers.

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I presume it to mean that an equation which gives valid or verifiable predictions within a certain range, once that range is exceeded, produces incorrect statements or values

This is exactly correct. But it does not then follow that there is anything at all wrong with the equation.

All equations, theorems, etc come with conditions of applicability or validity when we apply maths to physics (or other disciplines). That is to be expected because the maths is only a model of the other discipline, it is not the same thing. Obviously we choose the maths that accurately models our area of interest. It is a common failing to ignore these conditions, often with disasterous results.

A simple example would be the calculation of stresses and deflections in beams.

The 'normal' equations assume that the span is greater than 20 times the depth.

Short or deep beams that do not meet this criterion require the use of a different formula.

In such a beam, the standard formula will offer a perfectly possible stress that can indeed be generated by a (slightly) different load, but will not be the actual stress experienced by the deep beam. The standard formula will, however offer the correct stress in a standard beam.

In such circumstances we say that the standard equation 'breaks down' in the case of short or deep beams.

A more spectacular example of breakdown along the lines of ydoaPs' infinity would be the equation of specific energy in a travelling fluid. This becomes infinite at a discontinuity such as a hydraulic jump. This is typical of any equation that has a 1/(x-a) term which therefore involves division by zero at x=a.

Does this help?

Edited by studiot
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Or basic kinematics. KE=1/2 mv^2 fails to give accurate results when v approaches c. Ohm's law breaks down when applied to certain electronics.

Many physics equations break down at some point.

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You're probably hearing something like this: Newtonian mechanics is our simplest, most intuitive picture of how the universe works. But the answers it yields are really just useful approximations that break down when we start talking about things moving very fast (we need Special Relativity), when we're near a really strong gravitational field (we need General Relativity), and when we're describing phenomena at the atomic/subatomic level (we need Quantum Mechanics). Newtonian mechanics works very well for everyday stuff, but its equations simply don't apply to more extreme phenomena. I.e. the approximation breaks down.

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This is exactly correct. But it does not then follow that there is anything at all wrong with the equation.

All equations, theorems, etc come with conditions of applicability or validity when we apply maths to physics (or other disciplines). That is to be expected because the maths is only a model of the other discipline, it is not the same thing. Obviously we choose the maths that accurately models our area of interest. It is a common failing to ignore these conditions, often with disasterous results.

A simple example would be the calculation of stresses and deflections in beams.

The 'normal' equations assume that the span is greater than 20 times the depth.

Short or deep beams that do not meet this criterion require the use of a different formula.

In such a beam, the standard formula will offer a perfectly possible stress that can indeed be generated by a (slightly) different load, but will not be the actual stress experienced by the deep beam. The standard formula will, however offer the correct stress in a standard beam.

In such circumstances we say that the standard equation 'breaks down' in the case of short or deep beams.

A more spectacular example of breakdown along the lines of ydoaPs' infinity would be the equation of specific energy in a travelling fluid. This becomes infinite at a discontinuity such as a hydraulic jump. This is typical of any equation that has a 1/(x-a) term which therefore involves division by zero at x=a.

Does this help?

You've offered me some clarity on this. I would think that because mathematics are so involved as modelling tools in physics it's no surprise the two are frequently confounded. As my interest in physics is fairly recent I have to research much of this. I can see how a discontinuity in an equation could easily act as a source of 'break down'.

Thanks again.

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