Hamilton's principle: why is the integrated difference of kinetic and potential energy minimized?

I present an intuitive answer to an often asked question: why is the integrated difference K-U between the kinetic and potential energy the quantity to be minimized in Hamilton's principle? Using elementary arguments, I map the problem of finding the path of a moving particle connecting two points to that of finding the minimum potential energy of a static string. The mapping implies that the configuration of a non--stretchable string of variable tension corresponds to the spatial path dictated by the Principle of Least Action; that of a stretchable string in space-time is the one dictated by Hamilton's principle. This correspondence provides the answer to the question above: while a downward force curves the trajectory of a particle in the (x,t) plane downward, an upward force of the same magnitude stretches the string to the same configuration x(t).Comment: 7 pages, 4 figures. Submitted to the American Journal of Physic

Dirichlet boundary value problem for Chern-Simons modified gravity

Chern-Simons modified gravity comprises the Einstein-Hilbert action and a higher-derivative interaction containing the Chern-Pontryagin density. We derive the analog of the Gibbons-Hawking-York boundary term required to render the Dirichlet boundary value problem well-defined. It turns out to be a boundary Chern-Simons action for the extrinsic curvature. We address applications to black hole thermodynamics.Comment: 4 pages, revtex4, v2: added Refs., made one statement stronger, added footnote and added paragraph on single field inflatio

From geodesics of the multipole solutions to the perturbed Kepler problem

A static and axisymmetric solution of the Einstein vacuum equations with a finite number of Relativistic Multipole Moments (RMM) is written in MSA coordinates up to certain order of approximation, and the structure of its metric components is explicitly shown. From the equation of equatorial geodesics we obtain the Binet equation for the orbits and it allows us to determine the gravitational potential that leads to the equivalent classical orbital equations of the perturbed Kepler problem. The relativistic corrections to Keplerian motion are provided by the different contributions of the RMM of the source starting from the Monopole (Schwarzschild correction). In particular, the perihelion precession of the orbit is calculated in terms of the quadrupole and 2$^4$-pole moments. Since the MSA coordinates generalize the Schwarzschild coordinates, the result obtained allows measurement of the relevance of the quadrupole moment in the first order correction to the perihelion frequency-shift

Is it possible to formulate least action principle for dissipative systems?

A longstanding open question in classical mechanics is to formulate the least action principle for dissipative systems. In this work, we give a general formulation of this principle by considering a whole conservative system including the damped moving body and its environment receiving the dissipated energy. This composite system has the conservative Hamiltonian $H=K_1+V_1+H_2$ where $K_1$ is the kinetic energy of the moving body, $V_1$ its potential energy and $H_2$ the energy of the environment. The Lagrangian can be derived by using the usual Legendre transformation $L=2K_1+2K_2-H$ where $K_2$ is the total kinetic energy of the environment. An equivalent expression of this Lagrangian is $L=K_1-V_1-E_d$ where $E_d$ is the energy dissipated by the friction from the moving body into the environment from the beginning of the motion. The usual variation calculus of least action leads to the correct equation of the damped motion. We also show that this general formulation is a natural consequence of the virtual work principle.Comment: 11 pages, no figur

The Maupertuis principle and canonical transformations of the extended phase space

We discuss some special classes of canonical transformations of the extended phase space, which relate integrable systems with a common Lagrangian submanifold. Various parametric forms of trajectories are associated with different integrals of motion, Lax equations, separated variables and action-angles variables. In this review we will discuss namely these induced transformations instead of the various parametric form of the geometric objects

On the exhaust of electromagnetic drive

Recent reports about propulsion without reaction mass have been met on one hand with enthusiasm and on the other hand with some doubts. Namely, closed metal cavities, when fueled with microwaves, have delivered thrust that could eventually maintain satellites on orbits using solar power. However, the measured thrust appears to be without any apparent exhaust. Thus the Law of Action-Reaction seems to have been violated. We consider the possibility that the exhaust is in a form that has so far escaped both experimental detection and theoretical attention. In the thruster's cavity microwaves interfere with each other and invariably some photons will also end up co-propagating with opposite phases. At the destructive interference electromagnetic fields cancel. However, the photons themselves do not vanish for nothing but continue in propagation. These photon pairs without net electromagnetic field do not reflect back from the metal walls but escape from the resonator. By this action momentum is lost from the cavity which, according to the conservation of momentum, gives rise to an equal and opposite reaction. We examine theoretical corollaries and practical concerns that follow from the paired-photon conclusion. (C) 2016 Author(s).Peer reviewe

Kant’s epigenesis: specificity and developmental constraints

In this paper, I argue that Kant adopted, throughout his career, a position that is much more akin to classical accounts of epigenesis, although he does reject the more radical forms of epigenesis proposed in his own time, and does make use of preformationist sounding terms. I argue that this is because Kant (1) thinks of what is pre-formed as a species, not an individual or a part of an individual; (2) has no qualm with the idea of a specific, teleological principle or force underlying generation, and conceives of germs and predispositions as specific constraints on such a principle or force. Neither of these conceptions of what is “preformed”, I argue, is in strict opposition to classical epigenesis. I further suggest that Kant’s lingering use of preformationist terminology is due to (1) his belief that this is required to account for the specificity of the specific generative force; (2) his resistance towards the unrestricted plasticity of the generative force in radical epigenesis, which violates species-fixism; and (3) his insistence on the internal, organic basis of developmental plasticity and variation within species. I conclude by suggesting that this terminological and interpretative peculiarity is partly due to a larger shift in the natural philosophical concerns surrounding the debate on epigenesis and preformation. Specifically, it is a sign that the original reasons for resisting epigenesis, namely its use of specific, teleological principles and its commitment to the natural production of biological structure, became less of a concern, whereas unrestricted plasticity and its undermining of fixism became a real issue, thereby also becoming the focal point of the debate

La figure de la terre: determinée par les observations de messieurs de Maupertuis, Clairaut, Camus, Le Monnier, de l'Académie Royale des Sciences, & de M. l'Abbé Outhier, correspondant de la même Académie, Accomgnés de M. Celsius, Professeur d'Astronomie à Upsal: Faites par ordre du Roi au cercle polaire

La figure de la terre: determinée par les observations de messieurs de Maupertuis, Clairaut, Camus, Le Monnier, de l'Académie Royale des Sciences, & de M. l'Abbé Outhier, correspondant de la même Académie, Accomgnés de M. Celsius, Professeur d'Astronomie à Upsal: Faites par ordre du Roi au cercle polair

Estimation of daily cloud‐free, snow‐covered areas from MODIS based on variational interpolation

[1] NASA's MODIS global snow-covered area (SCA) products are one of the mission's major objectives that frequently contain cloud hindrances, which degrade their practical usability. Many techniques have been developed to mitigate the problem but with no assurance of eliminating all of the clouds. An image-processing algorithm with its kernel based on the variational interpolation theorem is developed to automatically obtain cloud-free dynamic SCA maps from MODIS. Two cases consisting of "accumulation" and "melting" phases are processed and validated using observations at 121 ground-snow sensors over the Sierra Nevada Mountains in California. The results show that the algorithm cleared all the cloud hindrance over the period of study. In terms of accuracy, the retrieved cloud-free snow cover for the accumulation case had an average omission error of around 22.5% and average commission error of around 2.1%, as compared to all available ground sensors. These high percentages of errors basically came from the input data of Terra and Aqua, which had omission errors of 14.3% and 20.2% (and the commission errors of ∼0.5%), respectively. For the melting case, when there were fewer clouds and hence more sensors available, the errors of omission and commission between the algorithm and direct observations from Terra and Aqua were close to each other (5.7-5.0% for omission and 0% for commission). © 2012. American Geophysical Union. All Rights Reserved