Finding trajectories that minimize the action

We have used the variational principle to determine if a given trajectory is realizable. We can also use the variational principle to find trajectories. Given a set of trajectories that are specified by a finite number of parameters, we can search the parameter space looking for the trajectory in the set that best approximates the real trajectory by finding one that minimizes the action. By choosing a good set of approximating functions we can get arbitrarily close to the real trajectory.

— Structure and Interpretation of Classical Mechanics


We have used the variational principle to determine if a given trajectory is realizable.


— Me@2019-03-29 04:23:36 PM


Check if the action of that given trajectory is stationary or not.

— Me@2019-03-29 04:25:45 PM



2019.03.29 Friday (c) All rights reserved by ACHK

Quantum classical logic

Mixed states, 2 | Eigenstates 4


— This is my guess. —

If the position is indefinite, you can express it in terms of a pure quantum state[1] (of a superposition of position eigenstates);

if the quantum state is indefinite, you can express it in terms of a mixed state;

if the mixed state is indefinite, you can express it in terms of a “mixed mixed state”[2]; etc. until definite.

At that level, you can start to use classical logic.


If you cannot get certainty, you can get certain uncertainty.


[1]: Me@2019-03-21 11:08:59 PM: This line of not correct. The uncertainty may not be quantum uncertainty. It may be classical.

[2]: Me@2019-03-22 02:56:21 PM: This concept may be useless, because a so-called “mixed mixed state” is just another mixed state.

For example, the mixture of mixed states

\displaystyle{p |\psi_1 \rangle \langle \psi_1 | + (1-p) |\psi_2 \rangle \langle \psi_2 |}


\displaystyle{q |\phi_1 \rangle \langle \phi_1 | + (1-q) |\phi_2 \rangle \langle \phi_2 |}



\displaystyle{\begin{aligned}  &w \bigg[ p |\psi_1 \rangle \langle \psi_1 |+ (1-p) |\psi_2 \rangle \langle \psi_2 | \bigg] +  (1-w) \bigg[ q |\phi_1 \rangle \langle \phi_1 | + (1-q) |\phi_2 \rangle \langle \phi_1 | \bigg] \\  &= w p |\psi_1 \rangle \langle \psi_1 | + w (1-p) |\psi_2 \rangle \langle \psi_2 | +  (1-w) q |\phi_1 \rangle \langle \phi_1 | + (1-w) (1-q) |\phi_2 \rangle \langle \phi_1 | \\  \end{aligned}}

— This is my guess. —

— Me@2012.04.15



2019.03.22 Friday (c) All rights reserved by ACHK

Find one, organize two

Technical debt


dna113 1 day ago [-]

I recently needed an HDMI cord for a monitor and realized that my cord drawer was accruing technical debt.

Whenever I am done with a cord I just throw it in there… it gets all tangled up with all the others. When I inevitably need one of those cords I impatiently pull it out and it makes all the other cords more tangled.

Here I am needing an HDMI cable that won’t just come out easily, I have to pay off my past laziness. But I have choices/tradeoffs/opportunities here.

I can just hurry up and get the minimum untangled and get back to watching TV.

I could untangle all of them since untangling one of them will help me untangle the others and wrap and label them.

I could just untangle the minimum, but also throw a roll of tape and a marker in there and wrap and label all future cords that go into that drawer, eventually they’ll all be nicely wrapped up and well documented.


jolmg 1 day ago [-]

Wow. I never thought of clutter in the home as technical debt, but it’s as similar as you describe. That really makes me see home organization in a whole new light.

— Technical Debt Is Like Tetris

— Hacker News


Whenever you have to search for something, once you have found it, organize an additional thing.

— Me@2019-03-12 11:12:28 AM



2019.03.13 Wednesday (c) All rights reserved by ACHK

(反對)開夜車 2.4

























— Me@2019-03-18 04:47:58 PM



2019.03.18 Monday (c) All rights reserved by ACHK

Physical laws are low-energy approximations to reality, 1.2


When the temperature \displaystyle{T} is higher than the critical temperature \displaystyle{T_c}, point \displaystyle{O} is a local minimum. So when a particle is trapped at \displaystyle{O}, it is in static equilibrium.

However, when the temperature is lowered, the system changes to the lowest curve in the figure shown. As we can see, at the new state, the location \displaystyle{O} is no longer a minimum. Instead, it is a maximum.

So the particle is not in static equilibrium. Instead, it is in unstable equilibrium. In other words, even if the particle is displaced just a little bit, no matter how little, it falls to a state with a lower energy.

This process can be called symmetry-breaking.

This mechanical example is an analogy for illustrating the concepts of symmetry-breaking and phase transition.

— Me@2019-03-02 04:25:23 PM



2019.03.02 Saturday (c) All rights reserved by ACHK

Computing Actions

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


\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



2019.03.10 Sunday (c) All rights reserved by ACHK

PhD, 3.4

碩士 4.4 | On Keeping Your Soul, 2.2.4 | Release early. Release often, 3.4

這段改編自 2010 年 4 月 18 日的對話。

























— Me@2019-03-06 12:11:31 AM



2019.03.06 Wednesday (c) All rights reserved by ACHK

Generalized Coordinates

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



2019.03.01 Friday ACHK