Technical supplements
This page is for readers seeking a bit more depth. It presents some mathematical models and detailed information about the physiology of dance movement and connection. Some familiarity with simple physics and math is assumed. I should stress that the models presented here do not constitute science; I don't have any experimental evidence, beyond my own experience as a dancer, to back them up. Still, I think they provide some interesting food for thought. Let's be honest; this is some serious nerd stuff right here, and if it's not your thing to go into this much detail about an activity that is first and foremost about having fun, then it should at least give you a laugh. Either way, enjoy!
1. Elastic frame
'Frame' is the relationship between the shape of a dancer's upper body -- everything between his/her centre of mass and point(s) of connection with a partner -- and the force that is applied to the partner.
In physical terms, this can in principle be represented by a vector (force) field over the space of possible connection points between partners. There will also be an associated scalar potential field; see contour lines in figure S1.
The overall frame may reasonably be broken down into two components: 1) the lean of the torso (relative positions of 'upper centre'/shoulders and 'lower centre'/hips), which is controlled by the core muscles; 2) The extension of the arms, which is controlled by the back, shoulder, chest and arm muscles. The force/shape relationships for each of these may take many forms, such as linear/elastic or nonlinear. In the elastic case, the applied force varies linearly with the change in shape of the upper body, compared with its 'resting' position. When both the torso and the arms behave elastically, the overall frame behaves elastically (see figure S1). The story is more complex when either frame component takes a more complex, nonlinear relationship, as we shall see below.
In order to build the story gradually, we start by assuming an elastic frame (figure S1) for each dance partner.
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| Figure S1: Elastic connection. 'Frame' is a relationship between the shape of the upper body and the force that it is applying to a dance partner. This can reasonably be broken down into two components: 1) the lean of the torso (relative positions of 'upper centre'/shoulders and 'lower centre'/hips), which is controlled by the core muscles; 2) The extension of the arms, which is controlled by the back, shoulder, chest and arm muscles. The coloured contour lines shown represent the 'potential fields' around the shoulders due to the core muscles, and around the hand due to the upper body muscles; red lines represent greater potential energy stored in the frame compared with green lines. The functions describing the associated force fields can have many forms, such as linear/elastic or nonlinear. When each of these two frame components behaves elastically, such that the force applied varies linearly with the change in shape, the combination of the two components (the overall frame) also behaves elastically (see below). The dancer and frame may therefore be modelled simply as a mass on a spring in this case (with driving and damping forces to be added later). |
2. Elastic connection
In the case of the partnership, we break the overall connection down into the leader's frame and follower's frame, and consider how these two sections work together to connect what the follower's centre is doing to what the leader's centre is doing. It follows that in a detailed analysis, one may choose to break down the partnership into four sections: leader's torso, leader's arms, follower's arms, follower's torso. And, if we want to, we can continue further with the decomposition, breaking down each arm and torso into different muscle groups, then to individual muscles and all the way to individual muscle fibres. However, we need not go this far. All the interesting physics is captured in a simple two-section decomposition, as we now demonstrate. We consider here the overall connection between partners but the reader is encouraged to realise that the analysis applies equally well to the two sections of an individual dancer's frame.
Consider two masses, $m_1$ and $m_2$, each connected to springs with elastic constants, $k_1$ and $k_2$ respectively, which are themselves connected like this:
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| Figure S1: Simple elastic model of dance connection |
We will consider the question, how does the interaction between the two masses (dancers - we can think of mass 1 as leader and mass 2 as follower) depend on the properties of the two springs (dancers' frames)? First we show that, provided each of the two springs does indeed behave like a simple spring (its restoring force varies linearly with the length of stretch or compression from its rest length), the overall interaction between masses may be modelled as a single spring with an elastic constant that is a function of the two springs' elastic constants. This is, of course, just a demonstration of the linear superposition principle.
First assume that mass 1 is fixed and mass 2 is free to move along the $x$ axis, as shown in Figure S2. When mass 2 is displaced a distance $x$ by a force $F$, the lengths of springs 1 and 2 will respectively be changed by $x_1$ and $x_2$ (i.e. $x=x_1+x_2$).
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| Figure S2: Connected two-spring system may be modelled as a single-spring system with effective elastic constant, $k$. |
The force is the same on each of the two springs, so
$$F=-k_1x_1=-k_2x_2.\label{equalfs}$$
Now, we assert that the two springs may be modelled as a single spring with elastic constant, $k$:
$$\begin{eqnarray}F&=&-kx\nonumber\\&=&-k(x_1+x_2).\label{assert}\\\end{eqnarray}$$
Using \ref{equalfs}, we get
$$k_2=k\left(\frac{k_2}{k_1}+1\right),$$
from which it follows that
$$k=\left(k_1^{-1}+k_2^{-1}\right)^{-1}.$$
Applying Newton's second law, we obtain the equation of motion for mass 2:
$$\ddot x=-m_2^{-1}\left(k_1^{-1}+k_2^{-1}\right)^{-1}x.$$
The frequency of oscillation is
$$\omega=\sqrt{\frac{k_1k_2}{(k_1+k_2)m_2}}.$$
It's interesting to note that
$$x_1=\frac{k_2}{k_1}x_2,$$
meaning that when one of the elastic constants is much larger than the other, the stretch in one spring will be much larger than the other. In practical terms this means that if one partner has a very stiff frame and the other a very loose frame, the latter will need to extend her/his arms much further than the partner with the stiff frame. Intuitively, there will be some ratio of $k_2$ to $k_1$ above which connection will become impractical because one partner's arms won't be long enough to stretch as required. This is a first hint that matching the $k$ values of the two partners ('frame matching') might be useful.
Clearly this model is too simple to capture dance connection even in one dimension, because the system is not driven; i.e. The leader (mass 1) is not adding energy to the system. The model is extended to accommodate this in section ______.
Despite the model's simplicity, it demonstrates an important principle: Provided both partners control their frames to behave like springs, with a linear relationship between the deformation of their shape and the force they're applying to the other partner, the partnership overall will also have such a relationship between force and deformation. This is useful in a social dance because it allows unfamiliar partners to dance together and connect, provided each follows a simple rule. "If I hold up my end of the bargain and you hold up yours, we both know how the partnership will feel and this is going to work."
An interesting question is, can this also be true when either/both partner(s) has a frame with a more complex, nonlinear relationship between the deformation of their frame and the force that they are applying to their partner?
3. Nonlinear connection
Consider a case in which each dancer's frame behaves not like a spring but rather like a 'superspring', with force-displacement relation of the form,
$$F=-(k_1x+k_2x^2+k_3x^3+.... +k_nx^n)=-\sum_{i=1}^nk_ix^i,$$
where $F$ is the restoring force and $x$ is the length of stretch or compression ('displacement') relative to the rest length.
We are interested in the question, if each dancer's frame is described by a polynomial of order $n$, can the overall partnership be described by a polynomial of the same order? We have already shown that this is true for $n=1$ and it is obviously also true for $n=0$ (the sum of two constants is a constant). How about $n\geq2$? And what about when the partners have different values of $n$; for example, one partner obeys a linear equation ($n=1$) and the other a quadratic ($n=2$)?
We first address the case where both partners have $n=2$. With reference again to Figure S2, we can write,
$$F=-k_{11}x_1-k_{21}x_1^2=-k_{12}x_2-k_{22}x_2^2,$$
where the second digit in each k-subscript now labels the superspring/partner (1 for leader, 2 for follower). We are seeking to describe the overall partnership with an equation of the form,
$$\begin{align}F&=&-&k_1x-k_2x^2\nonumber\\ &=&-&k_1(x_1+x_2)-k_2(x_1+x_2)^2\nonumber\\&=&-&k_1(x_1+x_2)-k_2(x_1^2+{\color{red}{2x_1x_2}}+x_2^2).\label{cross}\end{align}$$
$$\begin{align}F&=&-&k_1x-k_2x^2\nonumber\\ &=&-&k_1(x_1+x_2)-k_2(x_1+x_2)^2\nonumber\\&=&-&k_1(x_1+x_2)-k_2(x_1^2+{\color{red}{2x_1x_2}}+x_2^2).\label{cross}\end{align}$$
The cross-term, in red, shows immediately that there is a problem. This makes it impossible to find expressions for $k_1$ and $k_2$ that are independent of $x_1$ and $x_2$. The upshot is that the combination of supersprings cannot be represented as a single effective superspring of the same polynomial order. Repeating this derivation for any order higher than $n=2$ will also give cross-terms in the equation equivalent to \ref{cross}, so those cases will fail also.
How about when one partner's frame has $n=1$ and the other, $n=2$? In this case,
$$F=-k_{11}x_1-k_{21}x_1^2=-k_{12}x_2.$$
(to be continued...)
Read more...
How about when one partner's frame has $n=1$ and the other, $n=2$? In this case,
$$F=-k_{11}x_1-k_{21}x_1^2=-k_{12}x_2.$$
(to be continued...)
