Discrete Hamilton-Jacobi Theory

We develop a discrete analogue of Hamilton-Jacobi theory in the framework of discrete Hamiltonian mechanics. The resulting discrete Hamilton-Jacobi equation is discrete only in time. We describe a discrete analogue of Jacobi's solution and also prove a discrete version of the geometric Hamilton-Jacobi theorem. The theory applied to discrete linear Hamiltonian systems yields the discrete Riccati equation as a special case of the discrete Hamilton-Jacobi equation. We also apply the theory to discrete optimal control problems, and recover some well-known results, such as the Bellman equation (discrete-time HJB equation) of dynamic programming and its relation to the costate variable in the Pontryagin maximum principle. This relationship between the discrete Hamilton-Jacobi equation and Bellman equation is exploited to derive a generalized form of the Bellman equation that has controls at internal stages.Comment: 26 pages, 2 figure

Evidence for a rapid decrease in Pluto's atmospheric pressure revealed by a stellar occultation in 2019

We report observations of a stellar occultation by Pluto on 2019 July 17. A single-chord high-speed (time resolution $= 2\,$s) photometry dataset was obtained with a CMOS camera mounted on the Tohoku University 60 cm telescope (Haleakala, Hawaii). The occultation light curve is satisfactorily fitted to an existing Pluto's atmospheric model. We find the lowest pressure value at a reference radius of $r = 1215~{\rm km}$ among those reported after 2012, indicating a possible rapid (approximately $21^{+4}_{-5} \%$ of the previous value) pressure drop between 2016 (the latest reported estimate) and 2019. However, this drop is detected at a $2.4\sigma$ level only and still requires confirmation from future observations. If real, this trend is opposite to the monotonic increase of Pluto's atmospheric pressure reported by previous studies. The observed decrease trend is possibly caused by ongoing ${\rm N_2}$ condensation processes in the Sputnik Planitia glacier associated with an orbitally driven decline of solar insolation, as predicted by previous theoretical models. However, the observed amplitude of the pressure decrease is larger than the model predictions.Comment: 7 pages, 3 figures, accepted for publication in Astronomy and Astrophysic