Principle of Least Action in Dance

• Principle of Least Action
• Clean Dance Mechanics
• Rise and Fall Example
• Physics Glossary

The Principle of Least Action states

\[ \delta \int (T - V)\, dt = 0,\]

where T is the kinetic energy and V is the potential energy i.e., the integral of \(T - V\) is stationary. It does not require \(T - V = 0\) at every instant.

Conservative pieces

• \(T\): kinetic energy (translation + rotation)
• \(V\): potential (e.g., gravitational (V_g) from COM height; small elastic storage V_e in tendons/fascia)

  Non-conservative losses (real humans!)

  We extend Lagrangian mechanics with Rayleigh dissipation:

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

Then the equations of motion become (Lagrange + Rayleigh):

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

where Q_ext are external inputs (e.g., leader micro-leads) and \(J^\top\lambda\) enforces connection constraints C(q,t)=0

Leaders vs Followers

• Leader: primarily the forcing term (time-shaped impulse/torque).
• Follower: solves a constrained least-action tracking problem (minimizes added acceleration/braking under the constraints).

Smoothness = low \(\mathcal{D}\). “Heavy” = high \(\mathcal{D}\).

Worked (simple) example: see Follower step with rise/fall.
Couple model: see Leader–Follower Framing.
Plain-language glossary: Dance Physics Rosetta Stone.