TRiPoD (Temporal Relationalism incorporating Principles of Dynamics)

Abstract

Temporal Relationalism is that there is no time for the universe as a whole at the primary level. Time emerges rather at a secondary level; one compelling idea for this is Mach's: that time is to be abstracted from change. Temporal Relationalism leads to, and better explains, the well-known Frozen Formalism Problem encountered in GR and other background-independent theories at the quantum level. Abstraction from change is then a type of emergent time resolution of this. Moreover, the Frozen Formalism Problem is but one of the many Problem of Time facets, which are notoriously interconnected. The current article concerns modifications of physical formalism which ensure that once Temporal Relationalism is resolved, it stays incorporated. At the classical level, this involves modifying much of the Principles of Dynamics. I first introduce the anti-Routhian to complete the Legendre square of Lagrangian, Hamiltonian and Routhian. I next pass from velocities $\dot{Q}\mbox{}^{A}$ to changes d$Q^{A}$. Then Lagrangians are supplanted by Jacobi arc elements, Euler--Lagrange equations by Jacobi--Mach ones, and momentum requires redefining but actions remain unchanged. A differential (d) version of the Hamiltonian is required, giving rise to a variant of the Dirac approach based on a d-almost Hamiltonian subcase of the d-anti Routhian. On the other hand, the forms of the constraints themselves, and of Hamilton--Jacobi theory, remain unaltered.