Daily Archives: January 12, 2025

Ex 1.32 Path functions and state functions, 2.3

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Structure and Interpretation of Classical Mechanics

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2. On the other hand, the path function \bar f[q] and the path function \bar \Gamma (\bar f) \circ \Gamma [q] are not necessarily the same. Explain.

3. Give examples where they are the same and where they are not the same.

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In other words, path q^{(m)} = 0 for all m>n. Or put it simply, q(t) is a polynomial.

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The second case is when the path function \bar f [q] requires no derivatives of q with order higher than n. For example:

Assume that the path is \displaystyle{q = \sin(t)}, the osculating path is

\displaystyle{\begin{aligned} &\mathcal{O} (t_0, q(t_0), v(t_0), a(t_0)))(t) \\ &= q_0 + v_0 (t-t_0) + \frac{1}{2} a_0 (t-t_0)^2 \\ \end{aligned}}

and \displaystyle{\bar f [q] = Dq}.

Then \displaystyle{\bar f [q](t = t_0) = \cos (t_0)}; and

\displaystyle{\begin{aligned} & \left. \bar \Gamma (\bar f) \circ \Gamma [q] \right|_{t_0} \\ &= \left. \bar \Gamma (\bar f) \circ \Gamma[\mathcal{O} (t,q,v,a)](t)\right|_{t_0} \\ &= \left. \bar \Gamma (\bar f) (t, q, v, a)\right|_{t_0} \\ &= \left[ v_0 + a_0 (t-t_0) \right]_{t_0} \\ &= \cos (t_0) \\ \end{aligned}}

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\displaystyle{\mathcal{O} (t,q,v, \cdots, q^{(n)})} == a taylor series with finite length

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However, if a path function \bar g requires a higher derivative which is not provided by \Gamma:

\displaystyle{q = \sin(t)}

\displaystyle{\begin{aligned} &\mathcal{O} (t_0, q(t_0), v(t_0), a(t_0)))(t) \\ &= q_0 + v_0 (t-t_0) + \frac{1}{2} a_0 (t-t_0)^2 \\ &= \sin t_0 + ( \cos t_0 ) (t-t_0) - \frac{1}{2} \sin t_0 (t-t_0)^2 \\ \end{aligned}}

\displaystyle{\bar g [q] = D^3q},

then \displaystyle{\bar g [q](t = t_0) = - \cos (t_0)}; and

\displaystyle{\begin{aligned} & \left. \bar \Gamma (\bar g) \circ \Gamma [q] \right|_{t_0} \\ &= \left. \bar \Gamma (\bar g) \circ \Gamma[\mathcal{O} (t,q,v,a)](t)\right|_{t_0} \\ &= \left. \bar \Gamma (\bar g) (t, q, v, a)\right|_{t_0} \\ &= \left[ 0 \right]_{t_0} \\ &= 0 \\ \end{aligned}}

— Me@2025-01-12 06:43:55 AM

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