How can Two particles in perpendicular S.H.M. interfere? While studying the simple harmonic motion of a particle, I came across this concept of the interference of two particles in S.H.M. Everything made sense until it was about two mutually perpendicular particles on the X and Y axis.
I don't understand how two particles performing their independent oscillations can affect a particle on the other axis (They do, take the Lissajous figures). I could be missing something fundamental, but this one has been plaguing me for so long. Please help.
Also, when I ask this question, I also wonder how two particles in S.H.M. of the same amplitude and frequency, but different phase angles, be at the same position at the same time for them to interact.
Well, this entire time, I am assuming that two particles must collide to interfere. Is this assumption correct? I am sorry for being so silly but I can't help it. I don't understand this one phenomenon.
 A: 
I don't understand how two particles performing their independent oscillations can affect a particle on the other axis (They do, take the Lissajous figures).

No they don't. As you say, the oscillations are independent. The fact that the 2D trajectory $\big(x(t),y(t)\big) = \big(\sin(at+\delta),\sin(bt)\big)$ is generally complicated doesn't mean that the horizontal motion affects the vertical motion or vice-versa.
If you look at the "shadow" of the trajectory projected onto the $x$-axis, then the shadow will follow the simple harmonic trajectory $x(t)=\sin(at+\delta)$. If you look at the shadow projected onto the $y$-axis, it will follow the simple harmonic trajectory $y(t) = \sin(bt)$.
A: It is not the interference of two particles.
What is described by the Lissajoux figures is a complex motion of a single particle. This motion can be described as the superposition of two simple harmonic motions along two perpendicular axes. Same as the projectile motion on a parabolic trajectory can be described as the superposition of two motions along the x and y axes.
