Coupled oscillators - mode and mode co-ordinates

[joriarty]

For this question I'm not going to introduce the particular problem I am working on, rather, I am merely wanting some explanation of a concept which I can't seem to find in any of my textbooks. I suspect the authors think it is just too obvious to bother explaining .

I'm revising for a test and have the full worked solutions for this problem in front of me. I can follow the mathematics, but not the reasoning behind it.

The question:

Two masses M₁ and M₂ are connected by springs as in my expertly drawn diagram attached.

Show that the amplitude of the displacement of the masses is described by expressions of the form:
$$\psi _{1}\left( t \right)=A_{0}\cos \omega _{+}t$$
$$\psi _{2}\left( t \right)=A_{0}\cos \omega _{+}t$$

My worked solutions now say:

Notice that when the system is in mode 1, the quantity (x₂ - x₁) is always zero, and (x₁ + x₂) varies harmonically. In mode 2 the reverse is true. Let us define a set of variables:

$$q_{1}=\sqrt{\frac{m}{2}}\left( \psi _{2}+\psi _{1} \right)\; -->\; \dot{q}_{1}=\sqrt{\frac{m}{2}}\left( \dot{\psi }_{2}+\dot{\psi }_{1} \right)$$
$$q_{2}=\sqrt{\frac{m}{2}}\left( \psi _{2}-\psi _{1} \right)\; -->\; \dot{q}_{2}=\sqrt{\frac{m}{2}}\left( \dot{\psi }_{2}-\dot{\psi }_{1} \right)$$

My question:

What exactly are q₁ and q₂, and why should these be equal to $$\sqrt{\frac{m}{2}}\left( \psi_{2}+\psi_{1} \right)$$ etc? Why $$\sqrt{\frac{m}{2}}$$? Is there a more specific name for this law that I could look up?

I hope my question is easily understandable! Thank you for your help.

(note: for the sets of equations relating q₁ and q₂ to m and x, there should be a "≡" sign rather than an "=" sign - for some reason my TEX formatting comes out with "8801;" rather than a "≡" sign. Odd.)

 

[joriarty]

Formulae now fixed. I hope. Sorry if I confused anyone while I was editing things