Worked Example: Waltz Forward Step with Rise & Fall (Follower tracking a lead)
Simplified setup
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)
- θ(t): small yaw/rotation of the frame
Generalized coordinates: q = [x, z, θ].
Assumptions:
- Mass m, vertical ankle/arch elasticity ~ spring k around a neutral height z₀
- Small rotational inertia I about vertical axis for θ
- Rayleigh (dissipation) captures foot friction and “tone” viscosity with coefficients
cx, cz, cθ
- External lead inputs from Leader are generalized forces: Fx(t), cz(t), τ(t)
Energies (conservative)
Kinetic:
\[
T = \frac{m (ẋ² + ż²)}{2} + ½ | θ̇²\]
Potential:
Lagrangian: 𝓛 = T − V
Dissipation (non-conservative)
Rayleigh function:
\[
R = \frac{c_x ẋ^2}{2} + \frac{c_z ż^2}{2} + \frac{c_θ θ̇^2}{2}\]
These terms penalize unnecessary velocity (slip, heaving, over-twisting, push-pull).
Lagrange–Rayleigh equations (with external lead)
For each coordinate qᵢ ∈ {x, z, θ}:
\[
\frac{d}{dt}(\frac{∂𝓛}{∂q_i}) − ∂𝓛/∂q_i + ∂R/∂q_i = Qᵢ^{(lead)}\]
Compute terms:
1) Travel x:
\[
\frac{∂𝓛}{∂ẋ} = m ẋ → \frac{d}{dt} = m ẍ\]
No explicit x potential:
\[
\frac{∂𝓛}{∂x} = 0 \]
and
\[
\frac{∂R}{∂ẋ} = c_x ẋ\]
Equation:
\($m ẍ + c_x ẋ = Fₓ(t)\)$
2) Vertical z (rise & fall):
\[
\frac{∂𝓛}{∂ż} = m ż → \frac{d}{dt} = m z\]
and because because the Lagrangian 𝓛 = T − V
\($\frac{∂𝓛}{∂z} = \frac{−∂V}{∂z} = −m g + k(z − z_o )\)\( \)\(\frac{∂R}{∂ż} = c_z ż\)\( Equation: \)\(m z̈ + c_z ż + k (z − z_o) + m g = F_z(t)\)\( 3) Yaw θ (frame rotation): \)\(\frac{∂𝓛}{∂θ̇} = I θ̇ → \frac{d}{dt} = I θ̈\)\( Since there is no explicit θ potential in this minimal model \)\(\frac{∂𝓛}{∂θ} = 0\)\( \)\(\frac{∂R}{∂θ̇} = c_θ θ̇\)$
Equation:
\($I θ̈ + c_θ θ̇ = τ(t)\)$
What these equations show by intuition
Travel (x):
\($m ẍ + c_x ẋ = Fₓ(t)\)$
- Smooth acceleration profiles (small, continuous ẍ) and low slip (small ẋ against friction) minimize effort.
- A “clean lead” looks like a gentle, time-shaped Fx(t), not spikes. The follower then tracks with minimal extra braking → less R (loss).
Rise & fall (z):
\($m z̈ + c_z ż + k (z − z₀) + m g = F_z(t)\)$
- Gravity (m g) always “leans” the equation downward; muscles must control descent → that cost shows up as czż (loss) unless timing exploits small elastic recoil k (which is unlikely).
- Efficient R&F: shape z(t) as a shallow, smooth arc so ż and z̈ stay modest; don’t “drop then catch” (that spikes ż, z̈ → bigger czż loss).
Frame yaw (θ):
I θ̈ + cθ θ̇ = τ(t)
- Over-tone (big cθ) makes following rotation feel heavy. Elastic tone (small cθ) lets the system rotate with less loss.
- Leader’s torque τ(t) should be time-shaped, not jerky.
Coaching corollaries (directly from the math)
1) Shape impulses → Reduce loss
- Time your leads (Fx, Fz, τ) as smooth envelopes, not hits. Followers then need less braking → lowers R.
2) Glide, don’t grind
- Keep ẋ continuous; avoid micro stops/starts that spike friction cₓ ẋ. Foot pressure should feel adhesive, not sticky.
3) Rise & fall as a “soft spring”
- Treat ankles/arch like a mild spring around z₀. Don’t free-fall (big ż, z̈ → costly), don’t heave (fights m g). Aim for shallow, pendulum-like arcs.
4) Elastic tone beats rigid tone
- High cθ (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 (R). The best following solves the path with minimal added acceleration.
A tiny numerical thought experiment (sanity)
- Assume m = 60 kg, k = 800 N/m (very mild ankle/arch elasticity), cz = 120 N·s/m (viscous vertical control).
- A 2 cm overshoot in z (z − z₀ = 0.02 m) adds k (z − z₀) ≈ 16 N to fight—tiny alone, but with ż spikes it multiplies loss via cz ż.
- Moral: the shape (small ż, smooth z̈) saves more energy than “bigger spring.” Timing > strength.