# Lagrangians in generalized coordinates

The function $\displaystyle{S_\chi}$ takes a coordinate path; the function $\displaystyle{\mathcal{S}}$ takes a configuration path.

\displaystyle{\begin{aligned} \mathcal{S} [\gamma] (t_1, t_2) &= \int_{t_1}^{t_2} \mathcal{L} \circ \mathcal{T} [\gamma] \\ S_\chi [q] (t_1, t_2) &= \int_{t_1}^{t_2} L_\chi \circ \Gamma [q] \\ \end{aligned}}

\displaystyle{\begin{aligned} \mathcal{S} [\gamma] (t_1, t_2) &= S_\chi [\chi \circ \gamma] (t_1, t_2) \\ \end{aligned}}

# Computing Actions

$\displaystyle{\texttt{literal-function}}$ makes a procedure that represents a function of one argument that has no known properties other than the given symbolic name.

The method of computing the action from the coordinate representation of a Lagrangian and a coordinate path does not depend on the coordinate system.

# Exercise 1.4. Lagrangian actions

For a free particle an appropriate Lagrangian is

\displaystyle{\begin{aligned} L(t,x,v) &= \frac{1}{2} m v^2 \\ \end{aligned}}

Suppose that $x$ is the constant-velocity straight-line path of a free particle, such that $x_a = x(t_a)$ and $x_b = x(t_b)$. Show that the action on the solution path is

\displaystyle{\begin{aligned} \frac{m}{2} \frac{(x_b - x_a)^2}{t_b - t_a} \\ \end{aligned}}

— Structure and Interpretation of Classical Mechanics

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\displaystyle{\begin{aligned} L(t,x,v) &= \frac{1}{2} m v^2 \\ \end{aligned}}

\displaystyle{\begin{aligned} S_\chi [\gamma] (t_1, t_2) &= \int_{t_1}^{t_2} L_\chi (t, q(t), Dq(t)) dt \\ &= \int_{t_2}^{t_1} \frac{1}{2} m v^2 dt \\ &= \frac{1}{2} m v^2 \int_{t_2}^{t_1} dt \\ &= \frac{1}{2} m v^2 (t_2 - t_1) \\ &= \frac{1}{2} m (\frac{x_2 - x_1}{t_2 - t_1})^2 (t_2 - t_1) \\ &= \frac{1}{2} m \frac{(x_2 - x_1)^2}{t_2 - t_1} \\ \end{aligned}}

— Me@2006-2008

— Me@2019-03-10 11:08:29 PM

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# Configuration Spaces

The set of all configurations of the system that can be assumed is called the configuration space of the system.

## Generalized Coordinates

1. In order to be able to talk about specific configurations we need to have a set of parameters that label the configurations. The parameters used to specify the configuration of the system are called the generalized coordinates.

2. The $\displaystyle{n}$-dimensional configuration space can be parameterized by choosing a coordinate function $\displaystyle{\chi}$ that maps elements of the configuration space to $n$-tuples of real numbers.

3. The motion of the system can be described by a configuration path $\displaystyle{\gamma}$ mapping time to configuration-space points.

4. Corresponding to the configuration path is a coordinate path $\displaystyle{q = \chi \circ \gamma}$ mapping time to tuples of generalized coordinates.

The function $\displaystyle{\Xi \chi}$ takes the coordinate-free local tuple $\displaystyle{( t, \gamma (t), \mathcal{D} \gamma (t), ... )}$ and gives a coordinate representation as a tuple of the time, the value of the coordinate path function at that time, and the values of as many derivatives of the coordinate path function as are needed.

\displaystyle{ \begin{aligned} \text{generalized coordinate representation} &= \Xi (\text{local tuple}) \\ (t, q(t), Dq(t), ...) &= \Xi_\chi (t, \gamma(t), \mathcal{D} \gamma(t), ...) \\ \end{aligned} }

\displaystyle{ \begin{aligned} \text{generalized coordinates} &= q \\ &= \chi \circ \gamma \\ \\ q(t) &= \chi(\gamma(t)) \\ \end{aligned} }

\displaystyle{ \begin{aligned} t &\to \gamma: \text{configuration path} \to \chi: \text{generalized coordinates} = q \\ \end{aligned} }

\displaystyle{ \begin{aligned} (t, q(t), Dq(t), ...) &= \Xi_\chi (t, \gamma(t), \mathcal{D} \gamma(t), ...) \\ \\ \Gamma[q](t) &= (t, q(t), Dq(t), ...) \\ \Gamma[q] &= \Xi_\chi \circ \mathcal{T}[\gamma] \\ \end{aligned} }

— 1.2 Configuration Spaces

— Structure and Interpretation of Classical Mechanics

— Me@2019-03-01 03:09:25 PM

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2019.03.01 Friday ACHK

# Path-distinguishing function, 2

$\displaystyle{\gamma(t)}$ = configuration path function

$\displaystyle{\mathcal{F} [\gamma]}$ = a function of time that measures some local property of the path

……….It may depend upon the value of the function $\displaystyle{\gamma}$ at that time

……….and the value of any derivatives of $\displaystyle{\gamma}$ at that time.

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We can decompose $\mathcal{F} [\gamma]$ into two parts:

1. a part that measures some property of a local description

and

2. a part that extracts a local description of the path from the path function.

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— 1.3 The Principle of Stationary Action

— Structure and Interpretation of Classical Mechanics

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1. The function that measures the local property of the system depends on the particular physical system;

2. the method of construction of a local description of a path from a path is the same for any system.

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\displaystyle{ \begin{aligned} \mathcal{F} [\gamma] &= \mathcal{L} \circ \mathcal{T}[\gamma] \\ \mathcal{T} [\gamma] &= (t, \gamma (t), \mathcal{D} \gamma (t), ...) \end{aligned}}

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— 1.3 The Principle of Stationary Action

— Structure and Interpretation of Classical Mechanics

— Me@2019-02-22 11:46:50 PM

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2019.02.24 Sunday ACHK

# Path-distinguishing function

So we will try to arrange that the path-distinguishing function, constructed as an integral of a local property along the path, assumes a stationary value for any realizable path. Such a path-distinguishing function is traditionally called an action for the system. We use the word “action” to be consistent with common usage. Perhaps it would be clearer to continue to call it “path-distinguishing function,” but then it would be more difficult for others to know what we were talking about.

— 1.3 The Principle of Stationary Action

— Structure and Interpretation of Classical Mechanics

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2019.02.17 Sunday ACHK

# SICM

Structure and Interpretation of Classical Mechanics (SICM) is a classical mechanics textbook written by Gerald Jay Sussman and Jack Wisdom with Meinhard E. Mayer. The first edition was published by MIT Press in 2001, and [the] second edition was released in 2015. The book is used at the Massachusetts Institute of Technology to teach a class in advanced classical mechanics, starting with Lagrange’s equations and proceeding through canonical perturbation theory.

— Wikipedia on Structure and Interpretation of Classical Mechanics

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2019.02.02 Saturday ACHK

# 長頸豹 3

SICM, 3.2 | SICP, 2.2

（安：你上星期也有提及過，那本電腦界的神作。）

「靈感」中的「靈」，其實就是解「靈活」、「靈通」，即是有大量和多類型的資料或消息來源。

— Me@2014.06.03

# SICP, 3

Just as every day thoughts are expressed in natural language, and formal deductions are expressed in mathematical language, methodological thoughts are expressed in programming languages. A programming language is a method for communicating methods, not just a means for getting a computer to perform operations – programs are written for people to read as much as they are written for machines to execute.

— Lisp: A language for stratified design

— Harold Abelson, Gerald Jay Sussman

— SICP distilled

— jao

2013.05.31 Friday ACHK

# Hacker 3

.

The Jargon File has had a special role in acculturating hackers since its origins in the early 1970s. Many textbooks and some literary works shaped the academic hacker subculture; among the most influential are:

* Hackers: Heroes of the Computer Revolution, by Steven Levy
* Godel, Escher, Bach, by Douglas Hofstadter
* The Art of Computer Programming (TAOCP), by Donald Knuth
* The Mythical Man-Month, by Brooks
* Compilers: Principles, Techniques, and Tools (“the Dragon Book”), by Aho, Sethi, and Ullman
* Structure and Interpretation of Computer Programs (SICP), by Abelson and Sussman
* The C Programming Language (K&R), by Kernighan and Ritchie
* The Hitchhiker’s Guide to the Galaxy, by Douglas Adams
* The Tao of Programming, by Geoffrey James
* The Illuminatus! Trilogy, by Robert Shea and Robert Anton Wilson
* Principia Discordia, by Greg Hill and Kerry Thornley
* The Soul of a New Machine, by Tracy Kidder
* The Cuckoo’s Egg, by Cliff Stoll
* The Unix System, by Stephen R. Bourne
* Hackers & Painters, by Paul Graham
* The Cathedral and the Bazaar, by Eric S. Raymond
* The essays of Richard M. Stallman (many published in Free Software, Free Society: Select Essays of Richard M. Stallman)

– Wikipedia on Hacker (programmer subculture)

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.

[10] “Hacker” here means a highly skilled programmer, not a computer criminal. — Me

The basic difference is this: hackers build things, crackers break them. — Eric S. Raymond

In academia, a “hacker” is a person who follows a spirit of playful cleverness and enjoys programming.

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2010.03.04 Thursday ACHK

# SICM, 3.2

《SICM》（Structure and Interpretation of Classical Mechanics）中的編程語言，除了 Scheme 的本體外，還會用到作者特製的力學程式庫 Scmutils。而這個程式庫（library）卻只有 Linux 的版本，不能安裝在 Windows 之中。那導致我要特意在我的 Windows 中，先裝一個 virtual machine（虛擬機器），從而在那個虛擬機器之上，再安裝一個 Linux 作業系統。

— Me@2012.12.25

# 尋找時間的定義

SICM, 3 | SICP, 2

《SICP》跟《SICM》，同樣是超級名著。但是它不易理解。幸好，它官方網站提供了，一些 1986 年的講課錄影。當年（2006），尚未流行使用 YouTube，所以我要花很多時間，下載那些錄影。我還記得，每節課的錄影檔案，有 600MB 那麼大。

— Me@2012.12.23

# SICM, 2

《SICM》（Structure and Interpretation of Classical Mechanics）總共花了我兩年的時間。我大概由 2006 年，閱讀至 2008 年。除了閱讀課文外，我還把內裡的幾乎所有練習，無論是數學題，還是程式題，都一一擊破。要花那麼長的時間才能完成，主要是因為上班的工務繁忙。每日可以花在自修的時間，通常也不會超過半小時。

— Me@2012.12.21

# 程式員頭腦 14.3

SICM, 1.3

（安：那本書真的用 programming（電腦編程），來講解 mechanics（力學）？）

— Me@2012.12.18

# 概念滑轉

SICM, 1.2 | 程式員頭腦 14

“In almost all textbooks, even the best, this principle is presented so that it is impossible to understand.” (K. Jacobi, Lectures on Dynamics, 1842-1843). I have not chosen to break with tradition.

— V. I. Arnold, Mathematical Methods of Classical Mechanics, footnote, p. 246

Structure and Interpretation of Classical Mechanics

— Me@2012.12.17

# SICM

（安：我怎樣把 mechanics（力學）學得好一點？）

（安：經典力學。）

（安：暫時純粹為了求知和娛樂。）

— Me@2012.12.15

# Hacker 3

.

The Jargon File has had a special role in acculturating hackers since its origins in the early 1970s. Many textbooks and some literary works shaped the academic hacker subculture; among the most influential are:

* Hackers: Heroes of the Computer Revolution, by Steven Levy
* Godel, Escher, Bach, by Douglas Hofstadter
* The Art of Computer Programming (TAOCP), by Donald Knuth
* The Mythical Man-Month, by Brooks
* Compilers: Principles, Techniques, and Tools (“the Dragon Book”), by Aho, Sethi, and Ullman
* Structure and Interpretation of Computer Programs (SICP), by Abelson and Sussman
* The C Programming Language (K&R), by Kernighan and Ritchie
* The Hitchhiker’s Guide to the Galaxy, by Douglas Adams
* The Tao of Programming, by Geoffrey James
* The Illuminatus! Trilogy, by Robert Shea and Robert Anton Wilson
* Principia Discordia, by Greg Hill and Kerry Thornley
* The Soul of a New Machine, by Tracy Kidder
* The Cuckoo’s Egg, by Cliff Stoll
* The Unix System, by Stephen R. Bourne
* Hackers & Painters, by Paul Graham
* The Cathedral and the Bazaar, by Eric S. Raymond
* The essays of Richard M. Stallman (many published in Free Software, Free Society: Select Essays of Richard M. Stallman)

— Wikipedia on Hacker (programmer subculture)

.

.

[10] “Hacker” here means a highly skilled programmer, not a computer criminal. — Me

The basic difference is this: hackers build things, crackers break them. — Eric S. Raymond

In academia, a “hacker” is a person who follows a spirit of playful cleverness and enjoys programming.

2010.03.04 Thursday $ACHK$