While the Newtonian formalism works charmingly well for most classical problems, it is tedious to implement in complex systems involving multiple components as we have to keep track of the forces acting on each component. A classic example is the double pendulum. Solving it the Newtonian way involves keeping track of the forces exerted by one mass on the other and the tension on both the strings.
This looks quite challenging and any misstep in handling all of these equations would result in a disaster. Soon, we'll be exploring the Lagrangian Formalism that will make this problem much easier to solve. But before doing so, we need to understand the generalized coordinates.
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| Double Pendulum |
When we look at the double pendulum, the two masses, in principle, can move in all three dimensions of space. This gives us six degrees of freedom, 3 for each mass. But considering the fact that they are attached to two strings, we can restrict the motion of the masses to a plane.
Now the degrees of freedom reduces to 4. Further-more, we assume the strings do not stretch, this means that the length of string remains the same throughout the motion of the pendulum, restricting the motion of each mass within the arc of a circle with the radius equal to the length of the respective string. The motion of each mass can be tracked by the angle it makes with a line drawn passing through its mean position. We have essentially reduced the number of coordinates required to describe the system from 6 to 2.
The conditions that restrict the motion of the system are called the constraints. Here the constraints were the strings of fixed length to which the bobs were attached.
Let's try and formulate all of this in the form of equations.
We began assuming both the bobs were free to move in any direction. If $\vec{r}_1$ and $\vec{r}_2$ are the position vectors of $m_1$ and $m_2$, then
$$\begin{align}\vec{r}_1&= r(\vec{x}_1, \vec{y}_1, \vec{z}_1) \\ \vec{r}_2&= r(\vec{x}_2, \vec{y}_2, \vec{z}_2) \end{align}$$
The strings to which the bobs are attached will reduce their motion to the $x-y$ plane. Also, its rigidity ensures that the motion follows:
$$\begin{align}r_1^2 &= x_1^2 + y_1^2 \\ r_2^2 &= x_2^2 + y_2^2 \end{align}$$
Its much easier to move forward using polar coordinates.
Thus $\vec{r}_i$ is a function of $\theta_1$ and $\theta_2$
$$\vec{r}_i = \vec{r}_i(\theta_1, \theta_2)$$
$\theta_1$ and $\theta_2$ are the generalized coordinates.
We began with describing the system with the traditional coordinate system and then, introduced the various constraints in the form of equations. This reduced the motion of the system, or its degrees of freedom. We were then left with the minimum number of parameters that could completely describe the system. These are called the generalized coordinates. They are usually represented as the set $\{q_i\}$.
While these coordinates are not necessarily orthogonal to each other, they are linearly independent. Meaning, we can vary one of the coordinates in a continuous manner keeping the other coordinates fixed. This wasn't possible in the cartesian system. Fixing the $y$ coordinate meant that the $x$ coordinate could not take any value it pleased. But in the generalized coordinates, one could vary $\theta_1$ by fixing $\theta_2$, or the other way around, and either coordinates can take on any continuous value. This is possible because the constraints have already been accounted for in the generalized coordinates.
Constraints
In the above example, we had four constraint equations that helped us reduce the number of coordinates. There are many kinds of constraints that are classified as follows:
1. Holonomic Constraints: If the constraint can be written in the form of an equality, $f(\vec{r}_1, \vec{r}_2 \cdots, t) = 0$, then its called a Holonomic Constraint. Holonomic constraints are functions of coordinates and time. There are no velocity terms or differentials of the coordinates
Example: In a rigid body, the relative position of each particle with respect to the other is fixed. This is given by the constraint equation $(\vec{r}_i - \vec{r}_j)^2 = r_{ij}^2$
2. Non - holonomic Constraints: If the constraints cannot be expressed as an equality, then it is a non - holonomic constraint.
Example: A ball rolling down the surface of a sphere. We can constrain the position of the ball to be outside the sphere, but it is free to be anywhere beyond the surface of the sphere. In a two dimensional scenario, the constraint would be written as $x^2 + y^2 \geq R^2$ where $R$ is the radius of the sphere.
3. Rheonomic Constraints: Constraints that are time dependent are classified as Rheonomic Constraints.
4. Scleronomic Constraints: Constraints that are time independent are called Scleronomic Constraint.
The constraints we dealt with so far are holonomic constraints. The neat thing about these constraints is the fact that each holonomic constraint equation will reduce the degrees of freedom of the system by one.
So, if a system of $N$ particles are free to move in three dimensions, we'll have $3N$ generalized coordinates, 3 coordinates for each of the $N$ particles. This is denoted as
$$q = \{ q_1, q_2, q_3, \cdots, q_n \}$$
where $(q_1, q_2, q_3)$ corresponds to $(x_1, y_1, z_1)$ of the first particle, the next three generalized coordinates track the second particle, and so on. If $k$ holonomic constraints are introduced. Then the number of generalized coordinates is reduced to $3N -k$
A Note on the choice of generalized coordinates
While choosing the generalized coordinates, it is logical to choose them in a manner that they are perpendicular to the constraint forces. In the case of the double pendulum, the choice of $\theta$ was deliberate as it is always perpendicular to the Tension on the string. We'll learn the advantage of this while covering Virtual Work and deriving the D'Alembert Principle.
The Configuration Space
The generalized coordinates that describe a system span a space of their own called the Configuration space. The configuration space holds information of every possible state of a system. A particle moving in a one-dimensional space has a 1-D configuration space. A free particle on a plane has $\mathbb{R}^2$ as its configuration space. In general, the dimensions of the configuration space is equal to the number of generalized coordinates. Each point in this space represents a possible state of the system.
For example, a pendulum will have a configuration space which is a circle as it rotates about its point of suspension. Each point on the circle represents a snapshot of the pendulum in a certain position.
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| A system of two pendulums with 3 states |
Things get interesting when we consider a multi-particle system. Consider a system of two simple pendulums. To keep it simple, let's say that each pendulum has only three orientations as shown in the picture.
The configuration space of each pendulum individually consists of 3 points on a circle. But combined together, the system has nine possible states. We represent one of the pendulums by a large circle with three points, and the other with 3 smaller circles, each with 3 points indicating the 3 orientations. So, when the pendulums move in a continuous manner, the resulting configuration space is a torus.
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| The Configuration Space of the two-pendulum system |
This idea can be extended to a system of $N$ particles and $k$ constraints to obtain a configuration space of dimensions $n = 3N - k$. Now, a point in this $n$-dimensional space represents the state the system is in. The trajectory of the point with time tells us about the time evolution of the system.
References:
1. Classical Mechanics - Goldstein, Poole, Safko
2. Introductory Classical Mechanics - Morin
3. No Nonsense Classical Mechanics - Schwichtenberg
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