Clean Dance-Mechanics Framing (Leader + Follower + Losses)

Let

\[ q = [\,q_L,\, q_F\,]\]

be generalized coordinates for leader and follower (e.g., torso yaw, sway, COM height, foot angles). 'Losses' include friction, push-pull between the dancers, conflicting travel vectors (Tvec).

Lagrangian (conservative parts)

\[ \mathcal{L}(q,\dot q) = T_L + T_F - \Big(V_g^{(L)} + V_g^{(F)} + V_e^{(L)} + V_e^{(F)}\Big) - V_{\text{couple}}(q).\]

  • \(T\): kinetic energies
  • (Vg): gravitational PE (rise/fall COM height)
  • \(V<sub>e</sub>\): small elastic storage (tendons/fascia/arch)
  • \(V<sub>couple</sub>\) (optional): elastic handhold/frame geometry, partner offset

Dissipation (non-conservative drains)

\[ \mathcal{D}(q,\dot q) = \tfrac12\,c_{\text{floor}}\|\dot x_{\text{foot}}\|^2 + \tfrac12\,c_{\text{tone}}\|\dot\theta_{\text{frame}}\|^2 + \tfrac12\,c_{\text{drag}}\|\dot x_{\text{body}}\|^2 + \tfrac12\,c_{\text{couple}}\|\dot\Delta_{\text{handhold}}\|^2.\]

Think: foot friction, internal viscosity, over-tone, partner drag - all pure cost, pure energy loss.

Constraints (connection/hold)

Enforce handhold distance/orientation with

\[ C(q,t)=0,\]

entering via multipliers \($\lambda$\) (or by using \(V<sub>couple</sub>\) and/or a velocity penalty in \($\mathcal{D}$\)).

Equations of motion (Lagrange + Rayleigh)

\[ \frac{d}{dt}\!\left(\frac{\partial \mathcal{L}}{\partial \dot q}\right) - \frac{\partial \mathcal{L}}{\partial q} + \frac{\partial \mathcal{D}}{\partial \dot q} = Q_{\text{ext}} + J^\top \lambda.\]

Dancer-English

  • The system (both partners) seeks a path that minimizes wasted effort while satisfying connection.
  • The Leader shapes the constraints/reference.
  • The Follower solves a constrained least-action tracking problem (minimal extra acceleration/braking; use small elastic assists).
  • High \($\mathcal{D}$\) feels heavy; low \($\mathcal{D}$\) feels effortless.

See also: