Podcasts about hamilton jacobi

Donald Salisbury (Austin) gives a talk at the Workshop on the Problem of Time in Perspective (3-4 July, 2015) titled "The Intrinsic Hamilton-Jacobi Dynamics of General Relativity and its Implications for the Semi-Classical Emergence of Time". Abstract: The quantization of the general theory of relativity is notoriously difficult, in particular on account of the underlying general covariance and the consequent appearance of constraints in the classical Hamiltonian theory. The notion of time in the quantum theory is especially troubling since differing ideas of time suggest themselves depending on the quantum rules that are employed and the interpretations given to time in the classical theory itself. I will address the problem of time from a perspective in which constraints are implemented in a Hamilton-Jacobi framework through the use of intrinsic coordinates. The canonical approach is especially suited for this task. The decisive result is that the problem of time is even greater than one might have expected; there are arbitrarily many equally valid and possibly inequivalent time choices that one can introduce in this manner, all involving the use dynamical variables that are invariant under the action of the four-dimensional diffeomorphism-induced group as described in Pons, Salisbury, and Sundermeyer, Phys. Rev. (2009), 084015. I will review a Kuchař-inspired, but fully diffeomorphism covariant, classical Hamiltonian approach to general relativity in which spacetime scalar phase space variables are introduced that can serve as intrinsic coordinates. There corresponds to each choice a constraint which can be converted (as originally proposed by Asher Peres for conventional variables) into an Einstein-Hamilton-Jacobi (EHJ) equation. The choice of intrinsic coordinates is rendered simple in terms of these new variables, as are the choices in the new intrinsic EHJ equation. Indeed, the resulting intrinsic dynamics follows immediately from the EHJ equation. No Lagrangian is obtained, and this might be expected given that spacetime scalars must depend on time derivatives of the metric and their introduction into the Einstein action would result in the appearance higher derivative contributions. To each EHJ equation there corresponds a Wheeler-DeWitt quantum equation with its own emergent time. I will begin to examine possible quantum implications of the existence of distinct emergent intrinsic times.

Speaker: Prof. R. Vinter Abstract: Estimates on the distance of a nominal state trajectory from the set of state trajectories that are confined to a closed set have an important unifying role in optimal control theory. They can be used to establish non-degeneracy of optimality conditions such as the Pontryagin Maximum Principle, to show that the value function describing the sensitivity of the minimum cost to changes of the initial condition is characterized as a unique generalized solution to the Hamilton Jacobi equation, and for numerous other purposes. We discuss the validity of various presumed distance estimates and their implications, recent counter-examples illustrating some unexpected pathologies and pose some open questions.

Speaker: Prof. R. Vinter Abstract: Estimates on the distance of a nominal state trajectory from the set of state trajectories that are confined to a closed set have an important unifying role in optimal control theory. They can be used to establish non-degeneracy of optimality conditions such as the Pontryagin Maximum Principle, to show that the value function describing the sensitivity of the minimum cost to changes of the initial condition is characterized as a unique generalized solution to the Hamilton Jacobi equation, and for numerous other purposes. We discuss the validity of various presumed distance estimates and their implications, recent counter-examples illustrating some unexpected pathologies and pose some open questions.