IndisputableMonolith.Action.Hamiltonian
The Hamiltonian module supplies the standard mechanics Hamiltonian as the Legendre transform of the quadratic-limit Lagrangian derived from the J-cost action. Researchers recovering Newtonian trajectories from Recognition Science would cite it when moving from the Euler-Lagrange equation to phase-space dynamics. The module consists of direct definitions and one-line wrappers that apply the Legendre transform to expressions from QuadraticLimit and EulerLagrange.
claimThe standard Hamiltonian is $H(q,p)=p^2/(2m)+V(q)$, obtained as the Legendre transform $H=p·q̇-L$ of the Lagrangian $L(q,q̇)=½m q̇²-V(q)$ that arises in the small-strain regime of the J-action.
background
This module sits in the Action domain and completes the classical-mechanics recovery from the J-cost functional. Upstream, EulerLagrange shows that the cost-rate action S[γ]=∫J(γ(t))dt yields the EL equation J'(γ(t))=0, hence γ(t)≡1. QuadraticLimit then expands J(γ)≈½ε² for γ=1+ε with |ε|≪1, recovering the standard kinetic term ½m q̇². The Cost module supplies the base definition J(x)=(x+x^{-1})/2-1.
proof idea
This is a definition module, no proofs. It introduces standardHamiltonian, conjugateMomentum, hamiltonQDotEquation, hamiltonPDotEquation, totalEnergy and energy_conservation via direct definitions and one-line wrappers that apply the Legendre transform to the Lagrangian supplied by QuadraticLimit.
why it matters in Recognition Science
This module supplies the Hamiltonian required by EnergyConservationDomainCert and the Noether specialization in Action.Noether. It fills the classical limit step that converts the J-action into standard phase-space mechanics, enabling the domain certificates for Newton's second law and energy conservation along trajectories. It touches the quadratic approximation that bridges the phi-ladder to ordinary physics.
scope and limits
- Does not derive the Hamiltonian outside the small-strain quadratic regime.
- Does not address relativistic or quantum corrections to the Hamiltonian.
- Does not prove global energy conservation; that is deferred to the domain certificate.
- Does not include explicit numerical examples or trajectory integrations.