title: Examples of complete solvability of 2D classical superintegrable systems

authors: Chen, Yuxan

abstract: Classical (maximal) superintegrable systems in n dimensions are Hamiltonian systems with 2n - 1 independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system, without resorting to separation of variables techniques or trying to integrate Hamilton’s equations. We treat a family of 2nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler’s laws. Then we treat two 2nd order nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories, some of which are rather complicated. An interesting example is the occurrence of “metronome orbits”, trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post-Winternitz system, an example of a classical 4th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta. It is a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entrée to superintegrablity theory.

text: Examples of complete solvability of 2D classical superintegrable systems Classical (maximal) superintegrable systems in n dimensions are Hamiltonian systems with 2n - 1 independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system, without resorting to separation of variables techniques or trying to integrate Hamilton’s equations. We treat a family of 2nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler’s laws. Then we treat two 2nd order nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories, some of which are rather complicated. An interesting example is the occurrence of “metronome orbits”, trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post-Winternitz system, an example of a classical 4th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta. It is a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entrée to superintegrablity theory.

year: 2015

title: Intralinguistic motivation for pluralism about truth

authors: Ulatowski, Joseph W.

subjects: motivation

abstract: Critics of the scope problem that motivates pluralism about truth have argued that it is a pseudo-problem. If the criticism is correct, then truth pluralism is left unmotivated and potentially bankrupt. In this paper, I argue that closely related to the scope problem is another problem, which I call “the scalar problem.” If the property of truth is sensitive to how an agent expresses the truth predicate within a single linguistic discourse and different agents or groups of agents express truth differently within that discourse, then there are different ways of being true within the same linguistic discourse. Given this possibility, even if the scope problem fails, truth pluralism remains fully motivated.

text: Intralinguistic motivation for pluralism about truth Critics of the scope problem that motivates pluralism about truth have argued that it is a pseudo-problem. If the criticism is correct, then truth pluralism is left unmotivated and potentially bankrupt. In this paper, I argue that closely related to the scope problem is another problem, which I call “the scalar problem.” If the property of truth is sensitive to how an agent expresses the truth predicate within a single linguistic discourse and different agents or groups of agents express truth differently within that discourse, then there are different ways of being true within the same linguistic discourse. Given this possibility, even if the scope problem fails, truth pluralism remains fully motivated.

year: 2024

title: Race-ing and engendering the nation-state in Aotearoa/New Zealand

authors: Seuffert, Nan

subjects: Maori

abstract: During the past several decades the dynamics of law and policy reform in Aotearoa/New Zealand have prefigured and encapsulated contemporary trends in economic state restructuring, indigenous claims to refigure the nation, and gender struggles for political representation. Successive governments have restructured the economy and the state using radical forms of neo-liberal economic policy with the stated aim of globalizing the economy. Political action on the part of the indigenous Maori people, often led by Maori women, increased and diversified, and included demands for self-determination and redress of historical colonial injustices. In 1997, the conservative National Party’s Jenny Shipley become the country’s first white female Prime Minister, followed in 1999 by the election of a center-left coalition headed by Labour Party leader Helen Clark. This article analyzes the race and gender dynamics of these contemporary trends. First, it argues that policies of state structural adjustment and Maori political claims have converged in commodified settlements of past colonial injustices as corporate deals. These deals are temporary alliances of men across race to silence and erase the demands for self-determination or political power sharing of Maori activists, who are usually Maori women. Second, this article contends that the political instability created by rapid structural adjustment and the change to a Mixed Member Proportional (“MMP”) electoral system converges with white backlash against these settlements to open space for New Zealand’s first two female Prime Ministers. This article also suggests that the election of a second white female Prime Minister in New Zealand may reflect the (temporary) regendering of the role of Prime Minister to the feminine.

text: Race-ing and engendering the nation-state in Aotearoa/New Zealand During the past several decades the dynamics of law and policy reform in Aotearoa/New Zealand have prefigured and encapsulated contemporary trends in economic state restructuring, indigenous claims to refigure the nation, and gender struggles for political representation. Successive governments have restructured the economy and the state using radical forms of neo-liberal economic policy with the stated aim of globalizing the economy. Political action on the part of the indigenous Maori people, often led by Maori women, increased and diversified, and included demands for self-determination and redress of historical colonial injustices. In 1997, the conservative National Party’s Jenny Shipley become the country’s first white female Prime Minister, followed in 1999 by the election of a center-left coalition headed by Labour Party leader Helen Clark. This article analyzes the race and gender dynamics of these contemporary trends. First, it argues that policies of state structural adjustment and Maori political claims have converged in commodified settlements of past colonial injustices as corporate deals. These deals are temporary alliances of men across race to silence and erase the demands for self-determination or political power sharing of Maori activists, who are usually Maori women. Second, this article contends that the political instability created by rapid structural adjustment and the change to a Mixed Member Proportional (“MMP”) electoral system converges with white backlash against these settlements to open space for New Zealand’s first two female Prime Ministers. This article also suggests that the election of a second white female Prime Minister in New Zealand may reflect the (temporary) regendering of the role of Prime Minister to the feminine.

year: 2002-02-01