Cleonis
• Member for 9 years, 4 months
• Last seen this week
Stats
15,112
reputation
552k
reached
582
9
questions
Communities
View all

Link to the contact page on my website

Hamilton's stationary action

Currently my main interest is to spread transparent understanding of Hamilton's stationary action.

Hamilton's stationary action can be thought of as a machinery with internal moving parts.

The image below is a screenshot of an interactive diagram that is available on my website. The title of the page is 'Energy-Position equation'.

The purpose of the interactive diagram is to make the internal moving parts transparently visible, similar to how a transparent model of an engine makes the inner workings visible.

In a recent (oktober 8, 2021) answer to a question about the least action concept I took the opportunity to upload diagrams. The diagrams are in the form of animated GIF's, displaying response of energy values to sweeping out variation.

The black dots in the lower left sub-panel are points along a trial trajectory. Various sliders allow the trial trajectory to be changed in various ways.

The true trajectory is identified as follows:
The true trajectory has the property that the rate of change of kinetic energy matches the rate of change of potential energy over the entire length of the trial trajectory. The two energies are displayed in the upper left sub-panel

The curve for the potential energy has been flipped upside down to emphasize that the slopes of the curves line up: the green curve displays the minus potential energy.

The main slider, at the bottom, does a global variation sweep.

The dots in the lower right sub-panel display the value of the following three integrals:
integral of the kinetic energy (red)
integral of the minus potential energy (green)
Integral of the sum of kinetic energy and minus potential energy (blue)

Motion of the dots in the lower right sub-panel represents motion in variation space. (The variational parameter $$p_v$$, the horizontal axis, is the value of the main slider.) The slope of the motion in variation space corresponds to the derivative of Hamilton's action with respect to variation.

1
19
51
378
Score
171
Posts
29
Posts %
229
Score
85
Posts
14
Posts %
158
Score
62
Posts
10
Posts %
155
Score
70
Posts
12
Posts %
127
Score
53
Posts
9
Posts %
93
Score
60
Posts
10
Posts %
Top posts
53
Jun 13, 2021
50
Oct 31, 2020
37
Jun 10, 2021
24
Aug 22, 2021
24
Jul 7, 2018
23
Dec 2, 2021
21
Dec 26, 2012
18
Feb 20, 2021
Top Meta posts
3
0
Top network posts
View all network posts