Linear equations for amplitude factors

[aaaa202]

My book discusses the oscillations of systems for small pertubations around an equilibrium point. By considering taylors expansions of the potential and kinetic energy they are led to a set of equations for the amplitude factors. I have attached the crucial lines as a picture.
My question is to how they solve the system of equations. They say that we have n solutions so any nontrivial solution requires the determinant of the system of equations to be zero. I think I am misunderstanding something here, because how does that solve the system of equations? Everyone knows that a system has no solutions if the determinant is zero. So what information does requiring the determinant to be zero give?

 

[ehild]

My book discusses the oscillations of systems for small pertubations around an equilibrium point. By considering taylors expansions of the potential and kinetic energy they are led to a set of equations for the amplitude factors. I have attached the crucial lines as a picture.
My question is to how they solve the system of equations. They say that we have n solutions so any nontrivial solution requires the determinant of the system of equations to be zero. I think I am misunderstanding something here, because how does that solve the system of equations? Everyone knows that a system has no solutions if the determinant is zero. So what information does requiring the determinant to be zero give?

An inhomogeneous system of equations has no solution if the determinant is zero, but a homogeneous one has (nonzero) solution only when the
determinant is zero.

See the following equations:

A
x+y=0
2x+2y=0

B

x+y=0
x+2y=0which one has nonzero solution?

In case of small oscillation, you have an unknown parameter ω and you get the possible values of ω from the condition that the determinant is zero.

ehild

 

[voko]

A homogeneous linear system always has a trivial - all zeros - solution. This corresponds to the stationary solution of the original system of ODEs. But they are are looking for (non-stationary) perturbations in the vicinity of the stationary solution. So the linear system must have a non-trivial solution in that case. A non-trivial solution exists only if the determinant of the system if zero.

 

[ehild]

An inhomogeneous linear system always has a trivial - all zeros - solution.

You meant homogeneous linear system...

ehild

 

[voko]

You meant homogeneous linear system...

Thanks for pointing this out!

 

[aaaa202]

oh okay. I just always learned that a matrix is invertible if determinant≠0. So there exists one solution. What is the basic difference between that case for inhomogenous systems and this one, in which you somehow find solutions when the matrix is non-invertible?

 

[voko]

The basic difference is that (when det A = 0) the inhomogeneous system may have no solutions or may have infinitely many. The homogeneous system always has at least one solution (all zeros), and (when det A = 0) it has infinitely many non-zero solutions. You may want to read up on the theory of linear systems.