1.9 Abstraction of Path Functions

Structure and Interpretation of Classical Mechanics

Given a function $\displaystyle{f}$ of a local tuple, a corresponding path-dependent function $\displaystyle{\bar f[q]}$ is

$\displaystyle{\bar f[q] = f \circ \Gamma[q]}$ .

So while the input of $\displaystyle{f}$ is a tuple $\displaystyle{(t, q, v, \cdots)}$ , the input of $\displaystyle{\bar f}$ is an abstract path $\displaystyle{q}$ .

$\displaystyle{\begin{aligned} \Gamma [q] &= (t, q, v, \cdots) \\ \bar f &= f \circ \Gamma \\ \end{aligned}}$

Given $\displaystyle{\bar f}$ we can reconstitute $\displaystyle{f}$ by taking the argument of $\displaystyle{f}$ , which is a finite initial segment of a local tuple, constructing a path that has this local description, and finding the value of $\displaystyle{\bar f}$ for this path.

1. The goal is to use $\displaystyle{\bar f}$ to reconstitute $\displaystyle{f}$ .

2. The argument of $\displaystyle{f}$ is

$\displaystyle{\begin{aligned} \Gamma [q] &= (t, q, v, \cdots, q^{(n)}) \\ &= (t, q^{(0)}, q^{(1)}, \cdots, q^{(n)}) \\ \end{aligned}}$

3. Assume that we have the value of the initial position of the path, $\displaystyle{q_0 = q(t=0)}$ . Then the path $\displaystyle{q(t)}$ can be constructed by

$\displaystyle{q(t) = q_0 + v_0 t + \frac{1}{2} a_0 t^2 + ... +\frac{1}{n!} q^{(n)}_0 t^n}$

Note that while $\displaystyle{q}$ represents a path, $\displaystyle{q(t)}$ represents the coordinates of the particle location on the path at time $\displaystyle{t}$ .

Knowing the value of $\displaystyle{q(t)}$ for every moment $\displaystyle{t}$ is equivalent to knowing the path $\displaystyle{q}$ as a whole.

— Me@2023-12-19 08:16:40 PM

Physics Town

continues

1.9 Abstraction of Path Functions

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