title: Hamilton-Jacobi Theory and Superintegrable Systems

authors: Armstrong, Craig Keith

subjects: Hamilton-Jacobi theory

abstract: Hamilton-Jacobi theory provides a powerful method for extracting the equations of motion out of some given systems in classical mechanics. On occasion it allows some systems to be solved by the method of separation of variables. If a system with n degrees of freedom has 2n - 1 constants of the motion that are polynomial in the momenta, then that system is called superintegrable. Such a system can usually be solved in multiple coordinate systems if the constants of the motion are quadratic in the momenta. All superintegrable two dimensional Hamiltonians of the form H = (p_x)sup2 + (p_y)sup2 + V(x,y), with constants that are quadratic in the momenta were classified by Kalnins et al [5], and the coordinate systems in which they separate were found.

 We discuss Hamilton-Jacobi theory and its development from a classical viewpoint, as well as superintegrability. We then proceed to use the theory to find equations of motion for some of the superintegrable Hamiltonians from Kalnins et al [5]. We also discuss some of the properties of the Poisson algebra of those systems, and examine the orbits.

text: Hamilton-Jacobi Theory and Superintegrable Systems Hamilton-Jacobi theory provides a powerful method for extracting the equations of motion out of some given systems in classical mechanics. On occasion it allows some systems to be solved by the method of separation of variables. If a system with n degrees of freedom has 2n - 1 constants of the motion that are polynomial in the momenta, then that system is called superintegrable. Such a system can usually be solved in multiple coordinate systems if the constants of the motion are quadratic in the momenta. All superintegrable two dimensional Hamiltonians of the form H = (p_x)sup2 + (p_y)sup2 + V(x,y), with constants that are quadratic in the momenta were classified by Kalnins et al [5], and the coordinate systems in which they separate were found.

 We discuss Hamilton-Jacobi theory and its development from a classical viewpoint, as well as superintegrability. We then proceed to use the theory to find equations of motion for some of the superintegrable Hamiltonians from Kalnins et al [5]. We also discuss some of the properties of the Poisson algebra of those systems, and examine the orbits.

year: 2007

title: Measurement of tide induced changes to water table profiles in coarse and fine sand beaches along Pegasus Bay, Canterbury

authors: Ericksen, Neil

abstract: Measurements of changing water table profiles in beaches along Pegasus Bay, Canterbury, show an interchange of water between the sea and beach sand pores throughout a single semi-diurnal tidal cycle. The velocity of water escaping from the water table in response to an ebbing tide does not appear sufficient to elutriate material of silt size or larger from the beach. The low computed velocity is thought to be due to hydrostatic control, by sand dunes at the back of the beach, on water table amplitude. Fresh water and wave wash are considered important supplementary sources to that of tidal water in influencing water table profiles.

text: Measurement of tide induced changes to water table profiles in coarse and fine sand beaches along Pegasus Bay, Canterbury Measurements of changing water table profiles in beaches along Pegasus Bay, Canterbury, show an interchange of water between the sea and beach sand pores throughout a single semi-diurnal tidal cycle. The velocity of water escaping from the water table in response to an ebbing tide does not appear sufficient to elutriate material of silt size or larger from the beach. The low computed velocity is thought to be due to hydrostatic control, by sand dunes at the back of the beach, on water table amplitude. Fresh water and wave wash are considered important supplementary sources to that of tidal water in influencing water table profiles.

year: 1970

title: What’s in a name? The sense of reference, the rigidity of designators and the history of causes when determining the names of the two main islands of Aotearoa/New Zealand

authors: Parker, Martin

subjects: New Zealand

abstract: The New Zealand Geographic Board recently revealed that the two main islands of Aotearoa/New Zealand have never been officially named. The revelation made both national and international news and led to a large number of publicly aired comments on the matter, suggesting that something so fundamental and seemingly innocuous as naming is in fact likely to generate a good deal of controversy, no more so than in a bicultural/bilingual context. This paper will examine the phenomenon of naming from a number of angles. After discussing a layperson’s idea of what is meant by the meaning of names, we shall attempt to apply a number of theoretical models from the field of Language Philosophy to the case of the main islands of Aotearoa/New Zealand: the Descriptivist approach of Frege and Russell, Kripke’s Rigid Designators, and Causal Grounding theory. We shall conclude that, in the case of applying the first two of these, there is a question mark over whether any of the current names for the two islands, both Māori and Pākehā, are in fact names at all; in terms of applying the theory of Causal Grounding, we are drawn to the conclusion that in history lies the key to the future.

text: What’s in a name? The sense of reference, the rigidity of designators and the history of causes when determining the names of the two main islands of Aotearoa/New Zealand The New Zealand Geographic Board recently revealed that the two main islands of Aotearoa/New Zealand have never been officially named. The revelation made both national and international news and led to a large number of publicly aired comments on the matter, suggesting that something so fundamental and seemingly innocuous as naming is in fact likely to generate a good deal of controversy, no more so than in a bicultural/bilingual context. This paper will examine the phenomenon of naming from a number of angles. After discussing a layperson’s idea of what is meant by the meaning of names, we shall attempt to apply a number of theoretical models from the field of Language Philosophy to the case of the main islands of Aotearoa/New Zealand: the Descriptivist approach of Frege and Russell, Kripke’s Rigid Designators, and Causal Grounding theory. We shall conclude that, in the case of applying the first two of these, there is a question mark over whether any of the current names for the two islands, both Māori and Pākehā, are in fact names at all; in terms of applying the theory of Causal Grounding, we are drawn to the conclusion that in history lies the key to the future.

year: 2010-02