We’ll model ONE dancer (Follower) responding to the Leader’s cue using reduced coordinates:
\(x(t)\): travel along line of dance (LOD)
\(z(t)\): COM height (rise & fall)
\(\theta(t)\): small yaw/rotation of the frame
Generalized coordinates: \(q = [x, z, \theta]\)
Assumptions:
Mass \(m\), vertical ankle/arch elasticity ~ spring \(k\) around a neutral height \(z_0\)
Small rotational inertia \(I\) about vertical axis for \(\theta\)
Dissipation \(D\) captures foot friction and “tone” viscosity with coefficients \(c_x, c_z, c_\theta\)
External lead inputs from Leader are generalized forces: \(F_x(t), F_z(t), \tau(t)\)
Kinetic:
\[ T=m2(\dot{x}2+\dot{z}2)+I2 \dot{θ}2T = \dfrac{m}{2}(\dot x^2 + \dot z^2) + \dfrac{I}{2}\,\dot \theta^2T=2m(\dot{x}2+\dot{z}2)+2I\dot{θ}2\]
⚠️ Note on “potential energy”
In human motion, muscles do not store potential energy the way a spring or pendulum does. Gravity and elastic stiffness (ankle/arch) act as external forces, not recoverable energy stores.
For clarity, we therefore take the Lagrangian as purely kinetic:
\[ L=T\mathcal{L} = TL=T\]
and handle gravity + stiffness explicitly as generalized forces below.
Dissipation function (Rayleigh form):
\[ D=cx2 \dot{x}2+cz2 \dot{z}\,2+c\dot{θ}2 \dot{\dot{θ}}\,2\mathcal{D} = $$$$ \dfrac{c_x}{2}\,\dot x^2 + \dfrac{c_z}{2}\,\dot z^2 + \dfrac{c_\theta}{2}\,\dot \theta^2D$=$$ $$2cx\dot{x}\,2+2cz\dot{z}\,2+2c\dot{θ}\dot{\dot{θ}}2$$ These terms penalize unnecessary velocity (slip, heaving, over-twisting, push-pull). _Note: In classical mechanics this is often written as $R$. We use $\mathcal{D}$ for clarity, reserving “$V$” for true potentials (none here). --- ## Lagrange–Rayleigh equations (with external + field forces) For each coordinate $q_i \in {x, z, \theta}$\]
ddt (∂L∂\dot{q}i)−∂L∂qi+∂D∂\dot{q}i=
\[ \]
Qi(lead)+Qi(field).\frac{d}{dt}!\left(\frac{\partial \mathcal{L}}{\partial \dot {q_i}}\right) - \frac{\partial \mathcal{L}}{\partial q_i} + \frac{\partial \mathcal{D}}{\partial \dot {q_i}} =
\[ \]
Q_i^{(\text{lead})} + Q_i^{(\text{field})}.dtd(∂\dot{q}i∂L)−∂qi∂L+∂\dot{q}\,i∂D=Qi(lead)+Qi(field)
\[ - $Q^{(\text{lead})}$ = forces/torques from the Leader. - $Q^{(\text{field})}$ = gravity + stiffness effects applied directly. --- ### 1) Travel $x$ (line of dance):\]
∂L∂\dot{x}=m\dot{x} ⇒ ddt=m\ddot{x}\frac{\partial \mathcal{L}}{\partial \dot x} = m \dot x \;\;\Rightarrow\;\; \frac{d}{dt} = m \ddot x∂\dot{x}∂L=m\dot{x}⇒dtd=m\ddot{x}
\[ No explicit $x$-potential: $\dfrac{\partial \mathcal{L}}{\partial x} = 0$\]
∂D∂\dot{x}=cx\dot{x}\frac{\partial \mathcal{D}}{\partial \dot x} = c_x \dot x∂\dot{x}∂D=cx\dot{x}
\[ **Equation:**\]
m\,\ddot{x}\,c\,x\,\dot{x}=Fx(t)m\, \ddot{x} + c_x \dot{x} = F_x(t)m\ddot{x}cx\dot{x}=Fx(t)
\[ --- ### 2) Vertical $z$ (rise & fall):\]
∂L∂\dot{z}=m\dot{z} ⇒ ddt=m\ddot{z}\frac{\partial \mathcal{L}}{\partial \dot z} = m \dot z \;\;\Rightarrow\;\; \frac{d}{dt} = m \ddot z∂\dot{z}∂L=m\dot{z}⇒dtd=m\ddot{z}
\[ **Explicit external loads:** - Gravity: $Q^{(\text{field})}_z = -m g$ - Elastic ankle/arch: $Q^{(\text{field})}_z = -k(z - z_0)$ **Dissipation:**\]
∂D∂\dot{z}=cz\dot{z}\frac{\partial \mathcal{D}}{\partial \dot z} = c_z \dot z∂\dot{z}∂D=cz\dot{z}
\[ **Equation:**\]
m\ddot{z}+cz\dot{z}=Fz(t)−mg−k(z−z0)m \ddot z + c_z \dot z =
\[ \]
F_z(t) - m g - k(z - z_0)m\ddot{z}+cz\dot{z}=
\[ \]
Fz(t)−mg−k(z−z0)
\[ --- ### 3) Yaw $\theta$ (frame rotation):\]
∂L∂\dot{θ}=I\dot{θ} ⇒ ddt=I\ddot{\dot{θ}}\frac{\partial \mathcal{L}}{\partial \dot \theta} = I \dot \theta \;\;\Rightarrow\;\; \frac{d}{dt} = I \ddot \theta∂\dot{θ}∂L=I\dot{θ}⇒dtd=I\ddot{\dot{θ}}
\[ No explicit $\dot{θ}$-potential: $;\dfrac{\partial \mathcal{L}}{\partial \theta} = 0$\]
∂D∂\dot{θ}=c\dot{θ}\dot{θ}\frac{\partial \mathcal{D}}{\partial \dot \theta} = c_\theta \dot \theta∂\dot{θ}∂D=c\dot{θ}\dot{θ}
\[ Equation: $$I\ddot{θ}+c\dot{θ}\dot{θ}=τ(t)I \ddot \theta + c_\theta \dot \theta = \tau(t)I\ddot{θ}+c\dot{θ}\dot{θ}=τ(t)$$ --- ### ✅ Key Takeaways - We keep the **nice structure** of Lagrangian mechanics without pretending dancers have “stored PE.” - Gravity and stiffness show up honestly as _loads_ in the equations. - Losses are cleanly captured by $\mathcal{D}$. --- ## What these equations show by intuition **1) Travel ($x$):** $$m\ddot{x}+cx\dot{x}=Fx(t)m \ddot x + c_x \dot x = F_x(t)m\ddot{x}+cx\dot{x}=Fx(t)$$ - Smooth acceleration (small, continuous $\ddot x$) keeps effort low. - Frictional drag $c_x \dot x$ means jerky, stop–start motion bleeds energy. - A clean lead looks like a _shaped_ $F_x(t)$, not spikes — the follower then tracks with minimal braking. --- **2) Rise & Fall ($z$):** $$m\ddot{z}+cz\dot{z}=Fz(t)−mg−k(z−z0)m \ddot z + c_z \dot z = F_z(t) - m g - k(z - z_0)m\ddot{z}+cz\dot{z}=Fz(t)−mg−k(z−z0)$$ - Gravity ($m g$) always pulls downward; muscles must supply counter-force. - Arch/ankle stiffness ($k$) acts like a weak spring, but provides only a modest assist. - Vertical losses come from $c_z \dot z$: “dropping and catching” creates big $\dot z$ spikes and high cost. - Efficient R&F = shallow, smooth arcs; no free-fall, no heaving. --- **3) Frame Yaw ($\theta$):** - Torque $\tau(t)$ from the leader drives rotation. - Over-tone (large $c_\theta$) makes the frame feel heavy and resistant. - Elastic tone (moderate $c_\theta$) lets the system rotate with clarity and low drag. - Jerky torque inputs cost more than time-shaped ones. --- ### ✅ Core insight - Dancers aren’t storing “potential energy.” They are continuously balancing **forces** (gravity, stiffness) against **losses** (friction, viscous tone). - Efficiency comes from **shaping motion profiles** (smooth $\dot q$, gentle $\ddot q$), not from trying to “pump” energy like a pendulum. - In practice: glide → don’t grind, shape impulses → reduce loss, and keep tone elastic rather than rigid. ## Directly from the math 1. **Shape impulses → reduce loss** Time your leads ($F_x, F_z, \tau$) as smooth envelopes, not hits. Followers then need less braking → lowers dissipation $\mathcal{D}$. 2. **Glide, don’t grind** Keep $\dot x$ continuous; avoid micro stops/starts that spike friction ($c_x \dot x$). Foot pressure should feel adhesive, not sticky. 3. **Rise & fall as a “soft spring”** Treat ankles/arch like a mild spring around $z_0$. Don’t free-fall (big $\dot z, \ddot z$ → costly), don’t heave (fights $m g$). Aim for shallow, pendulum-like arcs. 4. **Elastic tone beats rigid tone** High $c_\theta$ (rigid frame) bleeds energy and makes rotation “heavy.” Enough tone to transmit intent; elastic enough to avoid viscous drag. 5. **Leader = reference; Follower = least-action solver** The best leads minimize follower dissipation ($\mathcal{D}$). The best following solves the path with minimal added acceleration. --- ## A tiny numerical thought experiment (sanity) - Assume $m = 60 \, \text{kg}$, $k = 800 \, \text{N/m}$ (very mild ankle/arch elasticity), $c_z = 120 \, \text{N·s/m}$ (viscous vertical control). - A 2 cm overshoot in $z - z_0 = 0.02 \,\text{m}$ adds $k(z - z_0) \approx 16 \,\text{N}$ to fight — tiny alone, but with $\dot z$ spikes it multiplies loss via $c_z \dot z$. - **Moral:** the *shape* (small $\dot z$, smooth $\ddot z$) saves more energy than “bigger spring.” $Timing > strength$. ---\]