Initial Conditions

Core Idea

Initial conditions select one trajectory from a family of solutions compatible with the same differential equation.

同一个 change law 可以允许许多 functions。initial conditions 给出某个起始时刻的 state，用来确定积分常数并选出具体 evolution。

对二阶方程 $\overset{y}{¨} = - g$ ，通常需要两个条件：

y (0) = y_{0}, \overset{y}{˙} (0) = v_{0} .

initial condition 不是附加在模型末尾的“小数据”，而是 state-space trajectory 的 starting point。对 pendulum，它通常是 $(θ_{0}, \dot{θ}_{0})$ 。

Differential Equations define the allowed motion.
Phase Space contains every possible initial state.
Euler Method begins from one initial state and steps forward.
Chaos can make tiny initial-condition errors grow into very different trajectories.
In indefinite integration, $f^{'} (x)$ determines an antiderivative family $f (x) = F (x) + C$ , while one value such as $f (x_{0}) = y_{0}$ determines $C$ : 第8课 · Initial condition and C.
Antiderivative