# How to visualize newtonian objects in term of harmonic oscillators?

From the group $$U(1)$$ in QED, via group representation we get the harmonic oscillators. However, I still have a hard time to imagine newtonian objects (computer, headphone, etc) as a combination of oscillators. I have take a look at classical limit but the visualization is still vague.

Let's say I have a glass breaking. How to visualize it in oscillations? • It is a really, really long trip. The rugged path is dubbed "classical limit". – Cosmas Zachos Feb 14 at 20:30
• @CosmasZachos can you elaborate? – Ooker Feb 15 at 10:06
• You may, painfully, discuss the classical limit, in phase space, of one oscillator; repackage trillions of these to classical field theory, reversing second quantization. Now, you claim you have an adequate classical field theoretic description of your phenomenon. Shortcut to models. Good luck. – Cosmas Zachos Feb 15 at 12:00

Suppose the object comprises $$n$$ degrees of freedom, and we work in a Cartesian coordinate system for these degrees of freedom, so each coordinate $$q_i$$ is $$0$$ at a stable equilibrium. The kinetic energy is of the form $$\sum_i\frac12m_i\dot{q}_i^2$$; the potential energy is of the form $$\sum_i\frac12k_iq_i^2$$, plus higher-order terms if an exact description of the system is anharmonic. (Cross terms, e.g. a term proportional to $$q_1q_2$$, can be deleted by rotating the coordinate axes.) Close enough to the equilibrium, such terms can be neglected. Such a harmonic approximation is of $$n$$ uncoupled harmonic oscillators.