---
title: Frameworks and Review
allDay: false
startTime: 12:30
endTime: 14:30
date: 2024-09-09
completed: null
type: single
---
# Introduction
Where do nonlinearities come from?
Well, a couple of places...
1. Geometric nonlinearities (pendulum)
2. External fields
3. Material properties
So we're stuck with them. But how do we deal with noninearities?

## A nonlinear equation
$$ \dot{x} = \frac{dx}{dt} = 1-2\cos x$$
How do you solve this? You can't use Laplace, you can't separate...
*insert very long expression that Bajaj wrote.*
Getting an analytical solution can be a PITA to obtain. For this reason:
**The general case is that nonlinear equations are unsolvable.**
This doesn't mean we can't learn things. We can describe these systems *qualitatively*.

Really our options come down to:
- Solve exactly (Not likely to happen)
- Solve numerically
- Analyze qualitatively (~geometrically)
- Solve an approximation to the problem
We mix and match these approaches. 
## Geometric (Qualitative) Methods