Federica wrote: ↑Wed Apr 23, 2025 4:41 pm
I haven't refreshed the pendulum example as a
non-conservation-of-energy example. What I meant is that the pendulum shows a case of
'inexplicable' incoming energy.
And the second bold implies precisely the first. My point was that in the case of the double pendulum, we observe behavior that seems unintuitive (like with the wingnut in space). Sometimes one arm of the pendulum seems to stop, as if it has exhausted its energy, then it abruptly starts moving again, as if it gathers energy from an inexplicable source. But this is really only a visual impression based on the fact that we are not accustomed to such behavior. A simple pendulum is easy to build an intuition for. We simply close our eyes and continue rhythmically swinging in our imagination. The double pendulum is not that easy. Sure, we can easily imagine it swinging wildly in our imagination, but the imaginary movements now seem willed more or less arbitrarily. If we return to the single pendulum, as we imagine the weight climbing up and slowing down, it feels very natural to bring it to a halt and smoothly accelerate the mental image back down. It is almost as if our intuitive curvature (etched by repeated observations of perceptual oscillatory processes) drags our imagination in its riverbed. When imagining a double pendulum, we feel more at loss. How do we decide if one arm should halt? How do we decide after how long it should abruptly start swinging again?
Nevertheless, the mathematical description of this process is very precise. Just search in youtube for "double pendulum simulation". It is a standard textbook exercise to derive the differential equations of motion of the double pendulum, and it is another standard task to make a numerical simulation. The reason why the movement seems unintuitive is because the potential energy of the system is defined in a more complicated way. What does this mean?
Imagine a classic example:
This is very intuitive to understand. Here the potential energy (PE) depends only on the height. The higher the cart is, the more potential for gaining speed (kinetic energy - KE) it has. Here's the place to remind that in physics, these energies should not be imagined as some sparkling 'energies', as some metaphysical bubbling substances. These are simply numerical assessments for the velocity and coordinates that have proven useful in practice. These numeric quantities can be thought of as currencies that can be exchanged. In the above image, we are low in the pit with high speed (higher KE) which can be 'exchanged' for height (higher PE). Then we reach the top but at the expense of losing speed. It is very different if we are low in the pit but also lacking velocity. Then we are simply stuck. We have no kinetic currency to exchange for height. Now we truly need an external source of energy, like the chain lift used for the initial climb of the rollercoasters. In this way, external energy is converted into height (PE) which then is converted into motion (KE) and happy screams.
We can do physics even without ever coming to the idea of KE and PE. We can simply apply the F=ma law at each step, and everything will build up just the same. Yet, thinking in terms of the accumulated motion (KE) and whether the configuration of the system is such that it can descend the potential gradient (PE) and gain motion, proves very useful, especially for more complicated systems. For rollercoasters and pendulums, this KE-PE exchange is as if almost implied in what we see. In the DP, however, there are two degrees of freedom. Thus, the PE doesn't depend only on the height of one weight but on two. These two degrees of freedom can be denoted with 𝜃
1 and 𝜃
2.
In other words, assuming that the masses and rod lengths are fixed, we need only these two angles to fully describe the state of the system. If we plot the PE of the whole system in the way it depends on both angles, we get something like this:
Here x and y correspond to 𝜃
1 and 𝜃
2, while the z height is the PE. Notice how in the middle, where both angles are zero we have the minimal PE. This is when both arms are pointed straight down (𝜃
1=𝜃
2=0). This corresponds to the motionless rollercoaster in the pit. The DP is stuck. If both arms are inclined, thus both 𝜃 are non-zero, we move toward the peaks at the corners. If only one angle is non-zero, the other zero, then we have intermediate cases along the x and y axes.
We need to switch our thinking here and realize that a point in this x, y space corresponds to the
state of the pendulum. That is, it is not a point in 3D space. This is the so-called configuration space or state space. Now imagine that we place a marble somewhere along this landscape. The x, y coordinates of that marble correspond to a specific state of the pendulum - specific 𝜃
1 and 𝜃
2 angles. What happens next? The marble is somewhere along the PE gradient. We let it go and it starts rolling in the direction of
steepest descent. The marble gains momentum. What does this motion correspond to? To the fact that the two angles of the DP are now changing. Some of the PE is exchanged for KE. Now we can let our imagination loose and picture how a marble would roll in such a landscape. It would accelerate downward but also gain speed. It then overshoots the lowest point and starts climbing against the PE gradient, losing KE in the process. We can easily see how complicated this rolling could be. For comparison, a simple pendulum can be imagined as the DP but with arms glued together such that 𝜃
1 is always equal to 𝜃
2. This would mean that marble is constrained about the diagonals (this is where x is always equal to y). If we try to imagine the cross-section of the landscape across these diagonals, we'll see a familiar valley (like the rollercoaster pit). Now everything is much more trivial. We can imagine the marble rolling in this valley back and forth, and this is the behavior of the simple pendulum.
The point to remember is that we're working in configuration space. The marble represents
a state of the DP, defined by the two angles. Yet, we can think of this space in terms of a potential landscape and the state of the system as the marble gaining or losing momentum.
We need to take note of two things. The gradient (the steepness) of the PE landscape is easy to determine. If a marble is placed at any point along this landscape at the moment of letting go it has zero KE because there's no movement yet through configuration space. Immediately, however, the PE gradient acts like a force that accelerates the system down the steepest direction.
Things are more complicated if we think of a marble that is already moving. Then it will also experience force along the steepest direction, but it still has inertia (KE) in some other direction. Thus, it will roll by inertia, but its direction will be bent toward the steepest direction. So we see that when we take the existing inertia and the PE bending force, it is more complicated to say what path the marble follows.
In numeric simulations this process is broken down in discrete steps. Let's say that the marble is at position x, y and it's current velocity is denoted with an arrow consisting of v
x and v
y components.
1. We calculate the steepness of the gradient (the force) at the current position
2. Every force yields acceleration (through F=m*a), and acceleration is a
change in the v arrow. Thus, the force will slightly turn the velocity arrow into the direction of greatest steepness, but not all the way.
3. We move the marble a little according to the v direction.
4. Repeat.
The precision of this method depends on how finely we slice the discrete steps. The finer they are, the more accurate. The coarser they are, the more the marble jumps from a position to a position and we update the arrow only after the jump. In other words, in between the two positions it is as if the force of the PE gradient doesn't exist. We ignore the PE field for a while and assume the marble travels as in free space, following the arrow. Then we momentarily reintroduce the PE field, push the arrow a little and repeat. Obviously, this introduces errors, the larger the time step is.
A more rigorous way to find the true path, which we would approach if the steps were to become infinitely fine, is achieved through the principle of least action. It gives us a method to, so to speak, filter out the path that is just right, where the changes of KE are consistent with the changes of PE.
The power of thinking in terms of KE and PE is that we can think about the whole system through such a marble analogy. Of course, most systems of interest have far more degrees of freedom than one or two. This would require that we visualize a multidimensional landscape. Mathematically, the gradient through the PE field is still trivial to calculate, and there's no problem to express movement as the changes of more than one, two, or three coordinates. Our visual intuition breaks down, but we can still conceive how the state of the system (the marble's coordinates in the multidimensional configuration space) continuously experiences force that tries to accelerate it 'down' the steepest gradient, while the marble still has its inertial motion.
All classical mechanics boils down to grasping this interplay. A DP if fully understandable in this way. It can be simulated with no problem. All of the bizareness of behavior is due to the more complicated PE landscape and we can get some intuitive sense for this if we imagine the marble rolling in the above shape. It is still not easy to imagine how every marble position corresponds to specific arm positions but we may gain some insight how the marble may slow down and change directions.
The reason our ordinary pendulum intuition breaks down is that we secretly try to see it through the prism of a simple marble rolling in a proper valley, where everything is simple and predictable. If we think in terms of the more complicated PE landscape, it is easier to see why the system exhibits seemingly unintuitive movements. However, there's nothing magical in this. There's no energy created or destroyed in ways that require us to imagine supersensible sources. It is all still an interplay between PE steepness and KE inertia.
This was a lengthy lecture but I just wanted to show that there's no inexplicable income of energy in the case of the DP or any other purely mechanical system. All confusion comes from us trying to fit the perceptions to our familiar intuition of simpler mechanics, like an oscillating weight on a spring or a rollercoaster. When there's a system with more degrees of freedom, these may look very unintuitive, but, as hopefully shown, in its essence, it's still a marble rolling along a gradient.
