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Buckethead
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I see comments such as "explains ... to the first order" or "to the second order" quite a bit in physics discussions. Can someone explain in lay terms, what first order and second order refer to?
Not quite. It is driven e.g. by the accuracy of measurements. It makes no sense to compute ##10## digits if you can only measure ##2##. Or it is given by the purpose. Earth's surface is curved, nevertheless we get along well with flat street maps. The error is just too small. But you better hope your pilot doesn't use flat directions on a trans-Atlantic flight.Buckethead said:Is it just guesswork as to when a solution is sufficient in the number of orders used?
Ideally yes, but first order approximations are linear approximations, i.e. tangents. Those are far easier to calculate and really often sufficiently close at small distances, cp. the examples above. They at least carry the tendency.For example I saw a discussion where an effect was not seen at first order but was seen at second order. So in general one should solve to as high an order as practical until you get diminishing returns?
No, that's a computational remark. E.g. solve ##x^5+c_1x^4+c_2x^3+c_3x^2+c_4x+c_5=0## can usually not be done via formulas, but only by numerical and thus approximation procedures. And there are a lot more problems, where we don't have closed forms for and only numerical approaches. The reason is that natural processes are often far more complicated than could be described by simple equations, so that algorithms will be necessary - and the computer has no number for ##\pi##, only an approximation.I also seen things such as "an exact solution could not be found" and I imagine this just refers to the order at which the equation was solved and does not refer to a failure at being able to solve the equation?
You have to remember that many of the mathematical models that are used to describe the Physical World do not involve simple polynomials. There are many models that involve trigonometrical and exponential functions etc etc.. When we talk about second (or other) order effects from these models, we are actually approximating them with a simple polynomial ( a quadratic or higher order).Buckethead said:Can someone explain in lay terms, what first order and second order refer to?
Thank you for that reminder. It is easy to forget that mathematical models of reality are just that and not the actual mechanisms by which reality operates and are therefore always going to be approximations. Excellent!sophiecentaur said:You have to remember that many of the mathematical models that are used to describe the Physical World do not involve simple polynomials.
In science, "first order" refers to a type of mathematical equation or model that describes a system or process in which the rate of change is directly proportional to the current state or amount of the system. This can also be referred to as a linear relationship.
The main difference between "first order" and "second order" in science is that while first order models have a linear relationship between the rate of change and the current state, second order models have a nonlinear relationship in which the rate of change is proportional to the square of the current state.
One example of a "first order" system in science is radioactive decay. The rate of decay of a radioactive substance is directly proportional to the amount of the substance present at any given time, making it a first order process.
In chemistry, the "order of reaction" refers to the relationship between the concentration of reactants and the rate of a chemical reaction. A "first order" reaction has a rate that is directly proportional to the concentration of one reactant, while a "second order" reaction has a rate that is proportional to the concentration of two reactants.
In data analysis, "first order" can refer to a type of regression model in which the relationship between two variables can be described by a straight line. This is often used to analyze trends and make predictions based on historical data.