Generalized Coordinates, Constraints, and D’Alembert’s Principle
Newtonian Mechanics and its Limitations
Classical mechanics, a topic that deals with the relationship between force, matter, and motion, has been around since the time of the ancient Greeks. The word “mechanics” itself is derived from the Greek root of μηχανική, which has a literal translation of “machines”. Aristotle and Archimedes were at the forefront of the development of classical mechanics, though it wasn’t until Newton put forth his laws of motion in 1687 before we truly had a way to connect force with motion.
The study of Newton’s laws and their consequences is termed Newtonian Mechanics, and in a large number of scenarios, it is seen to be particularly useful. Indeed, the Newtonian approach to solving a problem can be stated as a series of steps as given below:
- List all the forces acting on the system.
- Resolve these forces along coordinates axes to simplify calculations.
- Equate the resultant force along each direction to the rate of change of momentum in that direction. This will lead us to the equations of motion.
In most common scenarios, the above process isn’t even as bloated as the above steps make it out to be. Indeed, Newtonian mechanics makes solving situations like inclined plane motion a breeze.
However, there are a few limitations we encounter when we confine ourselves to the Newtonian approach.
- The general approach to solving scenarios in Newtonian mechanics involves Newton’s second law, i.e., the relation between momentum and force. This means that for our solution to be valid, we must have knowledge of the forces involved at all instants of times.
- The Newtonian process is geometrical, rather than analytical. The Newtonian approach involves vector quantities like the force, which must be resolved along our coordinate axes for the calculation to be reasonable.
- Newtonian Mechanics is easy to solve so long as the situations involved are idealistic. However, in practical situations, there are always restrictions on how our system can move, known in scientific terms as “constraints”. While the forces causing these restrictions do not appear directly in our outcome, they are still a nuisance to handle since they must be known beforehand while solving for the equations of motion.
Why The Lagrangian Approach is Superior
Lagrangian dynamics is just one of the many possible alternatives to the conventional Newtonian approach. It proves better suited for solving problems because of the following reasons:
- Much like its Hamiltonian counterpart, Lagrangian mechanics is analytical in nature and can even be generalized to quantum scenarios.
- Lagrangian mechanics deals in scalar quantities like energy, which means direction becomes insignificant.
- Lagrangian mechanics, as we will see, involves generalized coordinates, which are arrived at after simplifying the problem keeping the constraints in mind. This means some of the variables exit the picture even before we start our calculations.
Fundamental Concepts
Before one can dive into the Lagrangian approach, there are a few concepts that require fundamental understanding, and these are discussed below:
1. Constraints
A constraint is simply a restriction on the motion of the system.
This could be something as simple as saying that the particle must move in a straight line. On the other hand, constraints could also be painfully complex, though their general tendency is to make calculations simpler. This is because every additional, independent constraint reduces the number of independent coordinates required to specify the motion of the system.
Consider this as an example: in the Cartesian coordinates system, the independent coordinates required to specify the motion of the object are the x, y, and z components of its position. That is, it may be expressed, in general, as:
Now, I could constrain the motion of the object in two dimensions, which would be represented mathematically as z = 0. In such a case, each term in my calculations that involved z would now go to zero, leaving me with one less variable to work on.
A constraint could be represented mathematically in two distinct forms, leading to two types of constraints based on representation:
- Holonomic constraints: those that can be represented as equations. For instance, z = 0, x² + y² = 25, etc. In general, any constraint of the form f(r₀, r₁, r₂…) = 0 is said to be holonomic.
- Non-holonomic constraints: those that may not be represented as equations. Instead, they are written as inequalities. For example, a particle subject to the constraint that its position may not exceed a particular value, which we would represent as r ≤|a|.
There are other classifications based on time-dependence or energy conservation, but for now, this is all I shall disclose.
2. Generalized Coordinates
Generalized coordinates are a set of independent coordinates that can fully describe the motion of the system.
Here is how they differ from the coordinates you have studied about before: Imagine a particle constrained to move only along the circumference of a circle.
Naturally, for such a case, the cartesian system introduces naught but complexity. Instead, we use polar coordinates (r, θ, z). However, the particle’s distance from the center of the circle does not change. If we shift the origin to the center of the circle and align the z-axis perpendicular to the plane of the circle, then z = 0, and r = constant. Thus, we only need θ to fully describe the motion of the particle. Thus, in this case, θ is the generalized coordinate.
3. The Concept and Principle of Virtual Work
For any object that needs to be displaced in any random direction, some amount of time must be spent. That is, all practical, real displacements occur in a finite amount of time. No matter how small the displacement, it must take some time interval for the displacement to occur. However, the concept of virtual displacement is different.
A virtual displacement is an arbitrary and instantaneous displacement in the coordinates of the particle. It occurs without the slightest change in time.
The principle of virtual work states that: the virtual work of the applied forces is zero. The displacements considered while calculating the virtual work are any arbitrary displacement subject to the given constraints. In mathematical terms:
Naturally, you would ask the question “why is this significant?”. Indeed, my first reaction upon studying this topic was one of incredulity. Why would I need to ever understand this concept when it is painfully redundant?
Interestingly enough, discussing the virtual work principle isn’t limited to the realm of mechanics and some consider it “most general statement in the whole of Physics”. While I definitely do not possess the variety of knowledge required to confirm that claim, I can confirm that the virtual work principle is quite an essential tool. If nothing else, it allows us to equate the virtual work of any system to zero, giving us one additional equation to work with.
4. D’Alembert’s Principle
D’Alembert’s principle states that a particle is in equilibrium when subjected to a force F, minus an effective force that equals the rate of change of its momentum. Being in equilibrium, the virtual work done by such a force for any arbitrary displacement will be necessarily zero. That is,
On its face, D’Alembert’s Principle might seem like a redundant manipulation of Newton’s second law or an alternative representation of the same. However, the true implications of this principle are much more interesting to study. This is because using this principle, one can easily convert a problem statement of a dynamic system into a static problem.
The best part about D’Alembert’s Principle isn’t that it is ridiculously derivable from Newton’s second law, but rather the applications that it finds. In fact, this principle isn’t so different from the principle of virtual work and merely exists as an extension of the latter to serve the needs of dynamic cases. For the virtual work principle to be useful, our system must be in static equilibrium under the given forces. If it isn’t, we can apply D’Alembert’s principle to convert the problem to a static case and proceed from thereon.
Conclusion
I think this article has already droned on for far too long, and even then, it fails to capture the beauty that classical dynamics holds. Nevertheless, the reader now must have an idea about the limitations of Newtonian mechanics, as well as the ways in which the Lagrangian approach avoids them.
After discussing a few fundamental concepts that will come in useful for understanding Lagrangian dynamics, my next agenda will be to explain how to solve problems by using the Lagrangian L of a system. For that, keep your eyes peeled for the next article. Till then, don’t stop learning.
