Modeling Dynamics of Dissipative Systems

Thinking about Jack Wisdom's comment at the end of his MIT Earth, Atmospheric and Planetary Sciences Chaos and Climate talk on Chaotic Dynamics in the Solar System:

The tides on earth are coupled to the Solar System dynamics and I was wondering how to describe this mathematically. On Earth the tidal energy is dissipated by an array of more or less  synchronised rotational oscillations. See Dynamic Theory of Tides:

The result of all of this is that instead of a simple standing wave moving back and forth across the ocean, the tidal crest follows a circular pattern around the ocean basin, counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere. This is analogous to shaking a pan full of water in a circular manner, and watching the water follow a similar circular path as it sloshes around inside. This large scale circular rotation pattern of tides is called amphidromic circulation (Figure 11.2.4). The rotation occurs around a central amphidromic point or node, that shows little tidal variation, while the largest tidal ranges occur on the edges of the circulation pattern. In Figure 11.2.4 the amphidromic points are indicated by the dark blue areas where the white lines converge, like spokes from a bicycle wheel, and the dark red and brown areas show the regions of maximum tidal heights. The tidal maxima will rotate around the amphidromic points, taking about 12 hours for a complete rotation, leading to two high and two low tides per day in many places.

These models can be coupled with those of ocean currents, as described in this report by Lavelle et al dated November 1988: A MULTIPLY-CONNECTED CHANNEL MODEL OF TIDES AND TIDAL CURRENTS IN PUGET SOUND, WASHINGTON AND A COMPARISON WITH UPDATED OBSERVATIONS. In addition to this there are circulating eddies of warm and cold water that break off from the currents. See Helen Czerski - The Blue Machine.

I found this interesting dissertation by Chong Ai entitled Effect of Tidal Dissipation on the Motion of Celestial Bodies.

Tidal effects in celestial bodies manifest themselves in many ways. Tides cause periodic changes in sea and ground levels, they affect the length of day, and even volcanic activity. Tides cause effects on the scale larger than that of an individual body, affecting entire orbits of planets and moons.

In this thesis we focus on the effect of tides on the dynamics of orbits, leaving aside internal effects of tides on planets. This thesis addresses a gap in the literature. On the one hand, the mathematical theory of celestial mechanics is a classical subject going back to Newton, and it reached a high level of development by people like Legendre, Lagrange, Laplace, Jacobi, Poincaré, Moser, Arnold and others. Without exception (to our knowledge) this theory treats planets as point masses subject to Newtonian gravitational attraction, and without account for tidal effects. On the other hand, astronomers take more realistic models of the planets, but get few if any rigorous results. In this thesis we study problems which fall in the gap between these two approaches: they do include tidal dissipation on the one hand, making them more realistic than the classical system which completely ignores them, but we make this dissipation simple enough to be tractable mathematically.

To build dissipation into the equations of motion, we use the Routh method of introducing dissipation into Lagrangian equations of motion. According to this method, to write the equations of motion one only needs, in addition to the Lagrangian of the system, also the so–called Routh dissipation function: the power dissipated as a function of generalized coordinates and generalized velocities of the system. We choose a simple class of dissipation functions, leaving more general questions for future work

See Edward Routh. He was a kind of mathematical equivalent of a racehorse trainer. He also did some very nice tidying up of the course: Routhian Mechanics.

Here's a video about the Apache Point Lunar Laser-ranging operation. See APOLLO Apache Point:

The APOLLO (the Apache Point Observatory Lunar Laser-ranging Operation) system was originally developed by the APOLLO collaboration led by Tom Murphy at the University of California, San Diego, and made its first range measurements in 2005. It was the first lunar laser ranging system to achieve millimeter-level precision.

I guess they use a Caesium clock to time the delay between the emission of the pulse and the returned photons. See The Action Lab - Ulexite (How Does Television Stone Work?).

This observatory is part of a programme to try to measure the difference between "active" and "passive" gravitational mass. See How the Moon is helping us confirm Einstein’s relativity.

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Here's Jade's recent video on the Three Body Problem and Solar System Stability

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Sabine Hossenfelder on Chaos Control. It seems to work by training AI to abstract the syntax of a symbolic dynamics, and that gives the AI a very complex and, I imagine, highly non-linear model of the internal dynamics. See Brian Josephson on Organised Complexity and Thermodynamics for a relevant discussion.

Use these ideas to better understand Nature, not how to better control it, because control is always to some particular end, such as the production of cheap energy. But what are you going to do with that cheap energy? Power electric cars to take people to cities to do stupid jobs just for the sake of making a financial profit? How do you know that is a good thing to do in the long term? Instead you could, for example, look at how data on microclimates might give you better long-range forecasting, and help you understand the large-scale effects of local economic activities such as intensive agriculture to feed the people who spend their day in offices doing stupid things just to make a profit.

See also Some Interesting Things About Molecules.

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On Newton's thoughts on chaos control, see The Final Piece in the Solar System-Stability Puzzle? https://physics.aps.org/articles/v16/72

Newton thus wondered: Does the net effect of these periodically varying forces average to zero, so that the planets’ motions remain stable over long times, or is there a nonzero net value that causes the planets’ paths to change, potentially destabilising the system? Ultimately, Newton hedged his bets. He reasoned that the motion of the planets was unstable, and thus that the Solar System would occasionally fall apart. But he thought that when that happened, God would jump in and restore order, putting the planets back where they started. At the end of his book Opticks, the scientist writes, “…blind Fate could never make all the Planets move one and the same way in Orbs concentrick, some inconsiderable Irregularities excepted, which may have risen from the mutual Actions of Comets and Planets upon one another, and which will be apt to increase, till this System wants a Reformation.”