>[!warning] >This content has not been peer reviewed. # Variational principles — RST foundation Classical **variational principles** state that physical trajectories and fields are determined by **extrema of an action functional**: least action, minimum free energy, least resistance. In RST, laws that look like "optimal equations" are naturally read as **substrate dynamics that extremise a cost functional under constraints**. --- ## I. Classical side - **Hamilton's principle:** The physical path between two endpoints in configuration space makes the action \(S = \int L \, dt\) stationary (\(\delta S = 0\)), leading to the Euler–Lagrange equations. - **Least action and geodesics:** Many systems (e.g. light in optics, free particles in GR) can be recast as "paths of extremal action" or "shortest length" in an appropriate geometry. - **Other variational principles:** Minimum free energy in equilibrium statistical mechanics, minimum dissipation, and Bejan's **constructal law** ("evolving to provide easier access to currents") \([1–3]\). --- ## II. Mapping to RRT / RST | Variational concept | RRT / RST identity | Axiom / equation | |:---|:---|:---| | **Action functional \(S\)** | Effective **cost** of a history; integrated budget spent on maintaining identity and moving relations. | A2 (cost of resolution), A3 (translation). | | **Euler–Lagrange equations** | Stationarity conditions for optimal budget allocation under constraints. | RST field equation; resource triangle. | | **Least action path** | History that maximises average fidelity \(\mu\) for given endpoints and noise floor. | Identity loop; \(\mu(\eta, n)\). | | **Minimum free energy** | Special case where the cost functional is thermodynamic free energy. | Links to statistical mechanics / free energy. | | **Least resistance geometries** | Flow structures (backbones) that minimise cost per transported unit; compatible with \(n \approx 1.25\) constructal / SPARC dimension. | Constructal law; backbone dimension. | In this reading, RST laws are not arbitrary equations but **variational statements**: among all possible histories allowed by the Format and noise floor, the realised ones are those that **extremise a cost functional** that counts substrate work. --- ## Links - **Foundation index:** [[../Foundation index]] - **RRT axioms:** [[Relational Resolution Theory (RRT)]] - **Applications Roadmap:** [[../../Applications Roadmap]]