Normal Modes of Mechanical Systems - Oscillations and Instabilities

Give a brief introduction to finding normal modes from the potential energy surface of an N degree of freedom system, and the three types of modes mentioned above..
Example 2 degree of freedom system. Two masses connected by springs to walls and to each other. We analytically find the natural frequencies of the system near equilibrium and corresponding normal mode shapes..
General motion near equilibrium is made up of a sum of normal modes (via the superposition principle), which gives rise to Lissajous figures..
Geometric interpretation of the equations of motion near equilibrium in terms of a "force field" for the case of a positive-definite potential energy matrix. For the eigendirections, we have what looks like Hooke's law for a spring-mass system, and simple harmonic motion..
Normal coordinates: Using the normal modes as new generalized coordinates for the Lagrangian dynamics. The dynamics in the normal modes becomes decoupled and we consider the interpretation of quasiperiodic motion on tori parametrized by energy in each mode. For more information, see my video on "action-angle" variables https://youtu.be/z-dGZgq-6jg.
Why find normal modes? Because if have a mechanical system which is "forced", that is, has some oscillatory driving force that has a forcing frequency close to a natural frequency, we can get a large response, which could lead to failure. We illustrate this with a 1 degree of freedom model and show the frequency-response curve. For an extended discussion of the frequency response curves and resonance, see my video on "Frequency response curves for linear and nonlinear oscillators" https://youtu.be/eRnM5zVQsMc.