Euler-Lagrange Equation - what is it? (Part 1/2)

Anyone who's taken a slightly rigorous calculus course or learnt anything about differential equations knows the two names I've mentioned in the title. But perhaps you aren't aware about the equation or what it means.

Well, I've got this post to explain just that and answer these questions. But firstly, let me start off by stating why exactly this particular equation is important. A vital part of classical mechanics, the Euler-Lagrange equation is used a lot for optimisation problems. It's used mainly due to 2 large advantages:

1. Its form is essentially the same in any coordinate system. This means you can switch between cartesian, polar and other coordinate systems with multiple variables seamlessly using this equation. Trying to apply Newtonian equations in these different systems would be tedious without this 'translator'. In particular, this equation implicitly encompasses the basic laws of physics.

2. A fascinating solution to a part of this equation can reveal whether there is a conservation law.

I'll go over both of these in detail, although not all in one post, since that might get a bit too dense and boring. In this post I'm going to derive the actual equation and I'll go over its implications and advantages in my future posts. One big piece of feedback I received on my previous post was the lack of explanation and clarity in my notes. Taking some of this criticism in, I've decided to give a guided tour through my notes on this topic. Hopefully by the end of this post, most of you will walk away knowing more about this extremely useful and fascinating equation.

Before starting, I'd also like to mention that there are many approaches by which you can derive this equation. The approach I've used is a bit different from ones you might find in youtube videos. I've done this through my readings of Leonard Susskind's famous book "Classical Mechanics: The Theoretical Minimum". They're a great series of books from a great physicist.

So, let's get started. I'm gonna start with something very basic. Let's take two points:

How would you go from one point, the one on the left say, to the point on the right? Well, there are many routes you could take. One of the routes (and the fastest one) is a straight line. But you could also have wacky curves all around the page that then circle back to the point. But in a system consisting of these two points (let's think of the points as particles), when is the system stationary? in other words, when is this system at rest?

Well naturally, if we want a system to be a rest, the first assumption we make is that the net force in this system is 0 (F=0). Now, and this is key, we borrow from basic laws of physics and energy conservation. In these set of laws, it states that force (F) can be described as derivatives of potential energy (PE). So, if F=0 then:

From elementary calculus, we know that if the derivative of a function is 0, it describes minima, maxima or saddle points. Here's an illustration as a reminder:

We're looking at all of this as a particle, stationary in space, but what if the particle is moving (as it most often is)? How do you calculate stationarity then?

To solve this, let's first define a movement in x (position) by delta-x:

Our new equilibrium is when delta-v = 0, or when the potential energy does not move even as the particle does. This is going to be a new general law of equilibrium (or rather, of stationarity). If a particle is moving though, we are not very interested in different points where it existed (let's not get into their existence and probabilities waves - it's an exciting topic and one that I intend to cover in future posts but for now, lets stick to non-quantum mechanisms). Rather, we are more interested in the trajectory of the particle. If the existence of a particle is shown through a point in a coordinate system, the trajectory is naturally shown through functions. So now, we do the same things we did above, but now with functions.

Coming back to my original point, take the two points in the coordinate system I asked you about at the very beginning. Let's join these two points through a wavy curve:

I'm going to take two arbitrary points within this curve and try and find the trajectory of a particle running through these points (which are in yellow). Using Pythagoras' theorem, we derive the following equation:

After some boring algebra, we arrive at the following equation for s (the total distance between the two points). All I did in between these two steps is take a dx out of the square root and then integrate both sides to find s:

If s is the distance between the two points, minimising this will give us the shortest distance between the two points. Trying to find this, we venture further and get to this step:

Seems familiar right? Well, just take that whole integral term and replace it with v and its exactly what we had in our equilibrium condition earlier! Pretty neat, right?

This solution can be proven in any coordinate system. If you take two points on the surface of the Earth, you would still get the same equation, just a bit different because the shortest distance between the two points is no longer a straight line. But I'm not going to take up your time in this post any longer, I'll get to the point now.

Let's take a different coordinate system - one with time (t) and position (x) which is dependant on time. Take two points on the graph again and connect it with a curve.

Trying to find stationary points between these two time intervals, we use what we got before. Instead of dx, we now have dt. And instead of the whole square-root term, we have what we call the Lagrangian.

Hmm, that's surprising. Why is the Lagrangian a function of both position and velocity? Easy, distance speed and time have a relationship together.

If we wanted to make this stationary, we use our general equilibrium rule:

That seems complicated and difficult. Let's make it simpler by reducing it. This first thing to do is to divide the time interval into equal increments (factor of epsilon). Next, let's divide the trajectory into these division. Here is the result:

I've labelled 3 time increments on the graph: timepoints i-1, i and i+1. Calling back to an early integration class, we are going to do something very similar to Reimann sums. Instead of depicting the integral of dt, we are instead going to express this using summations. Specifically, if we focus in on the chosen time interval, we sum the Lagrangian over the time interval (i) to obtain the following equality:

The key to understanding this step is to understand the meaning of integral, which is really to find the areas underneath the graph. By summing these individual areas, we are obtaining essentially the same result.

To make this even simpler, let's only focus on the i time interval. Here's the graph above zoomed in:

From this, we have two terms, one for the distance between i-1 and i and one for the distance between i and i+1. Remember, we no longer have integrals but rather summations.

I've colour coded each term in a different colour. This is useful for our next step. We will now do exactly the same thing we did previously to find the equilibrium. We want to find where the derivative of this is 0. Taking the derivative of this term with respect to xi, we get:

Or, simplified:

So, after all of that, we get:

Or, to put it in a different way:

And voila! What you see above is the general form of the Euler-Lagrange equation.

In the second post, I'll go over how we derive Newtonian equations from this equation, how we can work in different coordinate planes seamlessly and how we can spot a conservation law. These topics are a bit more complex, but I'll try and break it down into simpler parts as much as possible!

I hope you enjoy this post! If you liked it, leave me a message! See you all soon!

What Newton did (and what he didn't do) - my notes on the principles of cosmology

The past few posts on this blog got a little bit off topic (I think at least). For a blog that's meant to be focused on astrophysics, it didn't really have much physics in it. So this week I decided to shift gears a little bit.

Here is not a blog post but rather an introduction towards a 2 part set of notes. These notes cover fundamentals from the principles of cosmology: Newtonian equations, Friedmann equations and finding a constant for the rate of expansion of the universe. All these derivations eventually lead to a model for the expansion of the universe over time.

These notes are heavily inspired and guided by the lectures of one of my idols, Leonard Susskind, conducted at Stanford University in 2013-2014. Here is a link to these lectures on YouTube: https://www.youtube.com/watch?v=P-medYaqVak

So, go ahead. Explore my notes on the origin of the theory of the origin. I'll be exploring the fundamentals of cosmology further in my future posts, but for now, let me know if these notes were helpful in any way, or even if they're so incomprehensible and muddled that I might as well have not have posted. I hope not. Good luck!

Lecture 1 - Newtonian equations and Friedman’s formula.pdf

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Lecture 2 - modelling the expansion of the universe with matter and radiation.pdf

Download PDF • 5.38MB

Rhea - The Mother of All Gods and Moons

The Moons of Saturn have proven to be an area of research with constant scientific interest. Rhea, one of the Moons existing in this system, is no ordinary astronomical body. From its size, the second largest Moon of Saturn, to its name, borrowed from Greek Mythology, Rhea signifies a body of aura and mystery in equal proportions.

Rhea was first discovered in 1672 by Giovanni Cassini whose achievements are of such magnitude that he possesses astronomical discoveries (Cassini Division) and space probes (Cassini Space Probe) as acknowledgments for his phenomenal work in astronomy. The body was the second satellite of Saturn discovered and the third object found to be orbiting the planet. Cassini had named this body, along with four other satellites of Saturn, “The Stars of Louis.” It was only in 1847 that English astronomer John Herschel suggested the name Rhea.

The early 21st century has proven to be pathbreaking in the study of Rhea. While the original hypothesis suggested that Rhea consisted of a rocky core, analysis of data from the Cassini flyby mission of 2005 proved otherwise. From these data, astronomers found the value for the standard gravitational parameter (GM), which is the product of the gravitational constant and the mass of the body, to be 153.93950.0018 km3s-2. From this, the density of Rhea was calculated to be 1232.85.4 kgm-3. Results for J2 and C22, degree-2 gravity coefficients, were found to be (7.9470.892)10-4 and (2.35260.0476)10-4. The ratio of these values suggests that Rhea has a hydrostatic equilibrium, which describes a balance between gravitational force, which works to contract an object, and pressure from the core, which works to expand an object. This equilibrium allows the body to maintain a spherical shape with no deviations from sphericity.

These gravity coefficients, along with the newfound conclusion that the body has a hydrostatic equilibrium, have also allowed the calculation of the moment of inertia, which is the radial distribution of mass in the satellite, using the formula: C/MR2to be 0.37210.0036 or approximately 0.4 which, theoretically, is characteristic of a spherical body with uniform density. (Iess et al.)

Through these analyses and calculations, astronomers continued to build sophisticated models for the interiors of Rhea. Most of such models describe an undifferentiated, or only weakly differentiated, satellite. These models have been backed by data from the flyby missions as well as the fact that the C22value is greater than the amount required for non-differentiated interiors to be existing (Castillo-Rogez.) The interior is further estimated to be composed of 25% rock and 75% ice mixtures.

These insights have been challenged over time. A paper from 2007 challenges the findings of a hydrostatic equilibrium and that the body is undifferentiated. By carrying out thermophysical dynamical evolution models of Rhea, this paper proposed that differentiation must have occurred near the origin of the body. The authors further go on to use their model to provide explanations of some non-hydrostatic anomalies that were observed by the Cassini mission. The paper suggests that the hypothesis of hydrostatic equilibrium was accepted a priori and therefore affected the results. Updated estimates for the gravity coefficients are also provided to back their claims. (Mackenzie et al.)

However, a different paper published in 2008 refuted these claims. The author of this paper suggests that the flyby mission returned “rather useless results” which gave no indication of the interior structure of Rhea. He goes on to describe how data from Doppler studies indicated that there was no convincing evidence that Rhea is nonhydrostatic (Anderson). Thus, the hypothesis of hydrostatic equilibrium was not rejected and the compositional makeup of the body (rock & ice) was found.

Emerging from the interior, the surface of Rhea, as is true for all the Moons of Saturn, is heavily cratered. Additionally, the planet is characterised by two large impact basins: Tirawa, which is upto 5 km deep and 360 km across, and Inktomi, with characteristics as interesting as its name. Known by its nickname ‘The Splat’, the basin is 47.2 km across and the impact that created it supposedly has ice from beneath Rhea’s surface. This phenomenon has resulted in the region appearing lighter than its surroundings. This light makes it extremely apparent in any image of Rhea, and is shown in Figure 1. Additionally, the impact also resulted in an ejecta blanket, which is a ring of ejecta surrounding a crater. Due to this observation, astronomers believe this to be a recent impact event.

Figure 1

What makes this Moon all the more intriguing is the fact that it may be the first of its kind to possess a ring system. This system is hypothesized to contain three dense, narrow bands within a disk surrounding the body.

In 2005, during Cassini’s flyby mission, the spacecraft had collected data from the plasma shadows (the regions where plasma from the magnetosphere of Saturn was blocked by the Moon) of Moons such as Dione and Tethys. The observations revealed an abrupt drop in energetic electrons, caused mainly by the blocking of the Moon. In Rhea’s observations however, the spacecraft found startling results.

There was a gradual decrease in the presence of energetic electrons in the distance leading up to the plasma shadow of Rhea. When the spacecraft entered the shadow, the expected cutoff was observed. But this distance over which electrons seemed to be decreasing in presence intrigued astronomers. Moreover, as the spacecraft exited from the shadow, a gradual increase in electrons was noticed, indicating a symmetric pattern of behaviour. To add to this symmetry, three sharp drops in electron presence were detected prior to entry and after exit from the plasma shadow. Further, the distance over which a gradual decrease and later increase of electrons was observed was corresponding to the Rhea’s hill sphere (the region around a body where any satellites are attracted into their orbit).

While initial study prompted gas and dust particles in the magnetosphere to be the likely candidates of such decreases, the spacecraft was not able to find these particles in large amounts. Therefore, the focus turned to larger, solid particles upto 1min size. These particles are predicted to have ‘soaked up’ the electrons forming a disk. This disk is said to be denser nearer to the Moon, which caused the drop in electrons, with the explanation for the three sharp anomalies being narrow disks of higher particle density (Hecht).

However, not all evidence supports this theory of a ring system. The primary source of objection is due to the fact that observations made by the Cassini ISS Narrow-Angle Camera failed to gather any evidence or material that can prove the existence of the ring system. A paper from 2010 rules out any solid objects or a disk-like structure to be the cause of the electron absorptions, essentially rejecting the hypothesis put forward by earlier studies (Tiscareno et al).

This paper further puts forward a theory that for any object to be absorbing these electrons, it must be four orders of magnitude larger than any particles observed currently. Specifically, this object must have an extinction coefficient, which can be written as r2nwhere r is the particle radius and n describes the density, more than four times that of any of the current observed particles.

The paper concludes by stating the codependent relationship between a particles’ mass and their abilities to absorb electrons. A large mistake made by earlier papers, according to the authors, was to consider the volume of a particle, rather than the mass, which would allow not just a ring of particles, but also clouds of matter around Rhea.

But, despite all this evidence, there’s an opening just wide enough for some astronomers to pursue this mystery further. Along the equator of Rhea, there are blue marks on the surface left by fresh blue ice which suggests an impact from deorbiting ring material. Figure 2 shows the blue marks on the surface.

Figure 2: Blue spots visible on the surface of Rhea, possibly from deorbiting ring material

Despite many astronomers rejecting the hypothesis of Rhea containing a ring system, the electron depletion pattern remains a reality, the cause of which is still largely unknown.

Energetic particles, such as the electrons discussed prior, also cause a different phenomenon in the atmosphere of Rhea. Cassini first detected Rhea’s exosphere and found oxygen and carbon dioxide molecules in it. Oxygen molecules originate from the energetic particles in Saturn’s magnetosphere. The particles collide with Rhea’s surface when Saturn’s magnetic field rotates over Rhea and cause chemical reactions releasing oxygen molecules as a byproduct.

This was not only the first time a space mission found oxygen molecules in an atmosphere other than Earths, but also suggests active chemistry occurring within Rhea’s atmosphere, a key ingredient for life on the satellite (Cook).

Life on the Moon is still far from reality, however.

Rhea’s temperature can reach upto 99K while in direct sunlight and range between 73K to 53K while in shade. Therefore, it is too cold on the surface for human life. And although we have made progress by identifying oxygen in the exosphere, this still does not translate to liquid water until the oxygen molecules are transported to a subsurface ocean, where life can be formed. No evidence of an ocean on Rhea has been detected till date.

The origin of the carbon dioxide still remains a mystery at large. Though there are many speculative theories, some backed by chemical knowledge, the true origin of these molecules is yet to be found.

Rhea continues to be a fascination and mystery for all those that study it. Possibilities of subsurface oceans, its largely unknown beginnings and the unapparent and possibly hidden system of rings make this body one of the more unexplored and mysterious satellites in our Solar System. Any further studies, particularly ones attempting to find the cause behind the depletion of electrons in the magnetosphere of Saturn and its link to Rhea, are particularly interesting and, according to me, require and deserve more attention.