D’Alembert’s helpful tips on easing your life
Well, last post was kind of a mess, sorry about that. That’s why you need planning in your subjects’ curricula (I looked it up, apparently this is the plural of curriculum). But I am not a teacher nor a professor, so I guess that gives me a pass!? (Give me a pass, please.) So, one of the things that got stuck in the air was these Euler-Lagrange Equations and Lagrangians and whatever. Most only ever come in contact with Newton’s mechanics, which is definitely enough for most pratical purposes, unless of course you are a engineer, or a physicist. And no I am not talking about quantum mechanics or special relativity. Even in our good ol’ classical mechanics, with our boring slow-moving, big, but not big enough bodies, sometimes Newton’s mechanics are not “enough”. And by not being enough, I don’t mean his laws don’t apply here. It’s just that for many cases, the problems in classical mechanics are too hard to solve with Newton’s mathematical approach. No new physics is required, just a change in the mathematics.
In the best way I can summarize the Newtonian approach as relating the motion of the system with the forces that are being applied on the system, with the motion of the particles in the system. Now that is simple to understand, very intuitive and revolutionary. Here’s the problem: If you have too many bodies interacting with each other, this becomes an incredible hard thing to do. And when you have constraints (these constraints are also forces in your system) which are usually complicated, it becomes a nightmare. Computers may be able to solve it, but back in the late 1700’s they had a lot of things, definitely not computers. And folks needed tools to deal with two things:
- First, it would be great if you could just get rid of coordinates that don’t really matter to your calculations, like, if you are in a room, and you want to locate another person in the same room, you don’t the vertical direction, right? Right?
- Second, getting rid of those constraints would be very helpful, we need freedom to move, to breathe, to live, and to do variational calculus.
Since, I’ve said that constraints are forces, that also means that the total resultant force F in the system is equal to the sum of all the non-constraint forces and the constraint forces, so we can break it down in the non-constraint parts, and the constraint parts.
If you don’t like words, we got you too, fam.
This image shows the constraint (C) and non-constraint (N) forces being applied on a particle. The first one shows a constraint that only allows one degree of freedom, the second shows a constraint with two degrees of freedom. Courtesy of Wiki
ALERT: Infinitesimals incoming! Beware of any infinitely small things that still manage to be slightly bigger than just zero.
I will try to avoid using a lot of mathematical lingo in this, but some of it will be unavoidable.
First, let’s just look at Newton’s laws of motion
Look’s very easy, right? And conceptually it is. But just assume you have a single particle. Since we leave in 3d space, that means solving the expression above 3 times, for each direction (up-down, front-back, left-right). And we usually solve for simple constant forces, that don’t change over time, giving us that typical quadratic equation (too much of a pain to write, just look for s = so + vt+ 0.5at(squared)). Now imagine a more complicated force, which makes that equation harder to solve. By the way, usually you may have many particle each with their individual 3 equations. And I didn’t even mention the constraints, which can make this even harder. In other words, better find an alternative.
Now, sure we may have found a way to break up forces into individual pieces, but how can that help us? And now, here is where things stop making physical sense. Virtual displacement!!! There you go, a fancy simple, but not clear word. What do they mean by virtual? By definition, virtual displacements are infinitesimal changes in the system, while t is constant. In a sense, it is a change in let’s say, the position of your particle by mantaining time constant. Now, here is a thing, nothing changes position without change in time. For something to change, time must not be constant. So you see that this virtual displacement is not something that makes sense. But again, this is why it is called virtual. These displacements are such that they reflect in some way the constraints of your system. And by putting in these virtual displacements, a French mathematician and physicist Jean le Rond D’Alembert (fancy) proposed (actually it was a Bernoulli, but do you have any idea how many of them were popular. I can’t keep track of all the Bernoullis around) and proved a principle, known now as the D’Alembert Principle, or Lagrange-D’Alembert Principle (because this is Lagrangian Mechanics, goddamit! I bet you if it was Newton, it would be Newton-D’Alembert Principle or something similar. You know how he was. Also because Lagrange wrote it first, but D’Alembert did a pretty big contribution to it) pretty much saying “You are not doing work at all“… Ups, I meant “ The total virtual work produced by the inertial forces (-ma) plus the impressed forces (applied forces) vanishes for reversible displacements ” (Again, thanks to Wiki for their ever welcoming free encyclopedia)
Quick side note: you know how linear momentum is defined as p = mv (v being the velocity of the particle or body)? So the derivative of the linear momentum dp/dt = vdm/dt + mdv/dt (btw, dv/dt is acceleration). Since for particles, and for most non-fluids, mass doesn’t change over time, the first term is equal to zero, and the second part is just the inertial force defined by Newton, so there you go, a new form to write Newton’s law: dp/dt = F. Or the change of momentum of a particle is equal to the force applied to that particle. This is important for what will follow.
Going back to the original point about D’Alembert’s principle, usually we first learn a more particular version of the principle, Hamilton’s Principle (Alexander Hamilton, Alexander Hamilton…) which applies only to specific systems where the constraints only depend on the positions and time (by the way, is this thing that has the complicated sounding name of holonomic system, because it has holonomic constraints, actually there are many kinds of holonomic systems, but I didn’t want to go over that rabbit hole), but I went with D’Alembert’s, that applies to any system.
The point is pretty much reiterated below, again for those who don’t like to read
By the way, the dot on top of p is to represent dp/dt. So, where was I? Right, to reach some conclusion about the D’Alembert’s principle. The linear product is between the virtual displacement and the sum between the applied forces and the inertial forces, for every particle in the system. Now, by definition, the constraints are not involved in the motion of the particle, but constrain it to a certain trajectory. That means that the motion will be perpendicular to the constraint applied. For the mathematically minded, this would just show up as:
So, while the constraints would limit the motion of the particle, it doesn’t affect the motion itself, the change in linear momentum, or the acceleration gained by the particle may be described just by the non-constraint forces applied to the system, which kinda makes sense, if you are being constrained from moving in a certain direction, you won’t be moving in that direction. Also, the principle is independent from the choice of coordinates. So this could apply whether you could only move in the left or right direction, or in any other direction you could conceive.
And while doing a bit of research on the topic, I ended up finding that there is an equivalent formulation, done by Gauss, that reaches the same conclusion but through a different approach, which is one of the cool things about mathematics, different folks have different strokes but they can all clean the same… uh, this didn’t come out as I wanted. Well, I hope you will have suggestions for these, and for how I can improve on the posts. And yeah, sorry about the nonlinear schedule, it’s just that I am terrible with time management, and that will come back to haunt me someday.
In this post, we used D’Alembert’s principle to rid our calculations from constraints, and now we’re free, baby. In another post, we will derive the Euler-Lagrange Equations from it, and hopefully, finish this saga.
Some useful resources
Goldstein, H – Classical Mechanics (This book is a classic one, and I have based many of my notations and deductions here from the book)