I want to ask about motivations of contact topology. I mean, for example, a symplectic manifold is a generalization of phase spaces and it explains why we need a closed, non-degenerate symplectic form.
I feel like a contact manifold also has the similar explaination, so we know why a contact structure should be a smooth maximally non-integrable hyperplane field.
Thanks in advance.
An example of motivation would be optics.
e.g. consider the 3-manifold $Y = \mathbb R^2 \times S^1$ (seen as as sub-bundle of $T\mathbb R^2$). Take coordinates $(x,y,\theta)$ on $Y$ and consider the contact structure $\xi$ on $Y$ given by $$ \xi|_{x,y,\theta} = \ker (\cos (\theta) d x + \sin (\theta) d y) $$ Then $\phi_t :Y\rightarrow Y$ given by $\phi_t(x,y,\theta) = (x+t\cos(\theta), y+t\sin(\theta), \theta)$ is a family of contactomorphisms. Now, if $\gamma$ is a Legendrian curve in $(Y,\xi)$, then $\phi_t\circ\gamma$ is also Legendrian at all $t$. Consider the projection $\pi : Y\rightarrow \mathbb R^2 ; (x,y,\theta)\mapsto (x,y)$. Then $\pi\circ \phi_t\circ\gamma$ can be seen as a wavefront evoluting in $\mathbb R^2$. This can be generalised to multidimensional wavefronts using the sphere-bundle inside $T\mathbb R^n$.
A similar example easily found online is about "why it's possible to park a car using contact geometry".
Another physical motivation would be pre-quantization "à la Souriau".