How do you solve systems of equations in complex physics problems?

[Fascheue]

It seems like systems of equation are part of almost every complex physics problem. What exactly are the rules or steps for solving such equations?

I know how to solve a basic linear system of equations with two variables, but how do solve more complicated systems like non-linear systems of equations with more than two variables? How do you know if you have enough equations to find a solution? I feel like I didn’t spend enough time in math on these sorts of things for the sorts of problems given in physics.

 

[fresh_42]

This is done on a case by case level and not answerable in this generality, except by some common remarks. In the end it is mostly a mixture of experience, try and error or templates of similar solutions. Systems of non linear equations are often only numerically solvable. There is an entire mathematical branch (numerical analysis) that deals with it. Also it depends on what you want to achieve: sometimes linear approximations will do.

 

[symbolipoint]

Any particular situation could be characterized using suitable systems of equations which may be non-linear. Mixture or blend problems might have a small number of linear equations with one involving a ratio. You would need a number of equations equal to the number of variables which you need solutions.

 

[Fascheue]

This is done on a case by case level and not answerable in this generality, except by some common remarks. In the end it is mostly a mixture of experience, try and error or templates of similar solutions. Systems of non linear equations are often only numerically solvable. There is an entire mathematical branch (numerical analysis) that deals with it. Also it depends on what you want to achieve: sometimes linear approximations will do.

Can systems of equations always be solved by substitution? And isn’t there some rule relating the number of unknowns to the number of equations?

 

[fresh_42]

Can systems of equations always be solved by substitution?

No, because one cannot always solve equations uniquely. $x^2=1$ has two solutions, and $\sin x = 0.5$ even infinitely many. Which one would you choose for a substitution? And what if the equation is $x^2+y^3+z^4=5$?

And isn’t there some rule relating the number of unknowns to the number of equations?

Not really. It comes from the substitution idea: $n$ equations with $n$ variables $x_1,\ldots ,x_n$ then write the equations as $x_i = \ldots$ and substitute them into the last, solve for $x_n$ and go backwards to get the other values. But as said above, this isn't always possible. Also some functions cannot be inverted to a closed form, e.g. $y=xe^x$ cannot be written as $x=\ldots$ Therefore this is more a rule of thumb: "Every equation decreases the degrees of freedom by one" than an exact theorem. Even in the case of linear equations you can have situations like $y=2x$ and $4x-2y=0$ which are formally two but effectively only one equation. And with more variables and equations, this situation is far less obvious than in my simple example.

Depending on the kind of equations and the task, there are sometimes theorems which can be used to find a solution. But often it can only be done by numerical algorithms. However, this adds a couple of more questions: how stable is a solution, i.e. if we use numbers, then we have to cut them at a certain point, $\sqrt{2}$ becomes $1.4142$, $\pi$ becomes $3.1416$, which aren't exact anymore. So what happens to the solution if we round differently? As I said, this is an entire mathematical branch. There is a famous theorem about differential equations, which states under which circumstances a solution exists and is unique. Unfortunately it doesn't tell how to find it. So as soon as linearity is left, things start to become complicated. That's one reason we often use linear approximations, because they are easy to handle. At least basically.

 

[FactChecker]

This is a good question. When you look into this, you will realize how much you have been spoiled by the beautiful math of the linear equations. Anything else can be a mess. That is why so often the first step of math, engineering, and physics is to try to linearize a problem.

 

[hilbert2]

One example where nonlinear systems of equations appear is the chemical equilibrium calculation for multistep reactions. For instance, you add ammonia (NH₃) to a copper sulfate solution of known concentration and the copper ions complexate stepwise with some number of ammonia molecules, forming [CuNH₃]²⁺, [Cu(NH₃)₂]²⁺, [Cu(NH₃)₃]²⁺ and [Cu(NH₃)₄]²⁺ species. If you try to calculate the relative amounts of these species at chemical equilibrium, you end up with a nonlinear system that has to be solved iteratively or with some approximations.