In many areas of math, I've been surprised at how much research, past and present, focuses on second order 'things'. Examples:
- Number theory: quadratic reciprocity, quadratic number fields
- Analysis: Second-order PDE's, Lagrangian equations, Newton's laws
- Geometry: conic sections, quadric surfaces, quadratic forms
- Topology: Surfaces
- Calculus: Second-derivative test
And many more. Introductory PDE classes and number theory classes may spend the entire semester on quadratic things.
Why is this so common? Is it because we can't handle third-order things, or is it because second order things are more important/common than higher-order things?