On Noether's theorem for the invariant of the time-dependent harmonic oscillator

Abe, Sumiyoshi; Itto, Yuichi; Matsunaga, Mamoru

doi:10.1088/0143-0807/30/6/011

Cited by 6 publications

(4 citation statements)

References 17 publications

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“…It has been applied to a wide variety of problems including quantum optics and related systems [13][14][15], non-stationary quantum field theory [16][17][18], geometric phases [19][20][21] quantum computation [22] and quantum cosmology [23]. It is worth emphasizing that the classical counterpart of the Lewis-Riesenfeld strong invariant can be derived by Noether's theorem [24] (see also [25] for a simplified discussion). Its generalization to dissipative systems has recently been presented in [11], where the weak invariant of the time-dependent quantum damped oscillator has explicitly been constructed by the use of the Lindblad equation.…”

Section: Introductionmentioning

confidence: 99%

Weak invariants in dissipative systems: action principle and Noether charge for kinetic theory

Abe

2020

Phil. Trans. R. Soc. A.

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In non-equilibrium classical thermostatistics, the state of a system may be described by not only dynamical/thermodynamical variables but also a kinetic distribution function. This ‘double structure’ bears some analogy with that in quantum thermodynamics, where both dynamical variables and the Hilbert space are involved. Recently, the concept of weak invariants has repeatedly been discussed in the context of quantum thermodynamics. A weak invariant is defined in such a way that its value changes in time but its expectation value is conserved under time evolution prescribed by a kinetic equation. Here, a new aspect of a weak invariant is revealed for the classical Fokker–Planck equation as an example of classical kinetic equations. The auxiliary field formalism is applied to the construction of the action for the kinetic equation. Then, it is shown that the auxiliary field is a weak invariant and is the Noether charge. The action is invariant under the transformation generated by the weak invariant. The result may shed light on possible roles of the symmetry principle in the kinetic descriptions of non-equilibrium systems. This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.

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Section: Introductionmentioning

confidence: 99%

Weak invariants in dissipative systems: action principle and Noether charge for kinetic theory

Abe

2020

Phil. Trans. R. Soc. A.

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“…Our experimental proposal focuses on the importance of exploring systems beyond the spring-mass and pendulum in order to boost the epistemic cognitive process. In this direction, electronics present an opportunity to introduce highly controllable damping and non-linearities to harmonic oscillators beyond mechanical systems [48,49] and, for advance courses, the ability to produce, for example, time-dependent control to introduce continuous symmetries and its invariants following Noether theorem [50]. We may look into space and introduce the damped harmonic oscillator using the dynamics of particular celestial bodies [51] or to more complex setups, for example, lasers and optical resonators [52], classical gases confined by harmonic potentials [53], or the oscillation of a superconductor ring levitated by a magnetic field [54].…”

Section: Literature Reviewmentioning

confidence: 99%

Effects of a Thermal Inversion Experiment on STEM Students Learning and Application of Damped Harmonic Motion

et al. 2021

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There are diverse teaching methodologies to promote both collaborative and individual work in undergraduate physics courses. However, few educational studies seek to understand how students learn and apply new knowledge through open-ended activities that require mathematical modeling and experimentation focused on environmental problems. Here, we propose a novel home experiment to simulate the dynamics of a flue gas under temperature inversion and model it as damped harmonic motion. After designing and conducting the experiment, twenty six first year students enrolled in STEM majors answered six qualitative questions to inform us about their epistemological beliefs regarding their learning process. Their answers imply that this type of open-ended experiments may facilitate students’ understanding of physical phenomena and point to the significance of physics instructors as promoters of epistemological development. In general, students described this activity as a positive experience that helped them connect an environmental phenomenon with a fundamental physics concept.

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“…Our experimental proposal focus on the importance of exploring systems beyond the spring-mass and pendulum in order to boost the epistemic cognitive process. In this direction, electronics present an opportunity to introduce highly controllable damping and non-linearities to harmonic oscillators beyond mechanical systems [47,48] and, for advance courses, the ability to produce, for example, time-dependent control to introduce continuous symmetries and its invariants following Noether theorem [49]. We may look into the space and introduce the damped harmonic oscillator using the dynamics of particular celestial bodies [50] or to more complex setups, for example, lasers and optical resonators [51], classical gases confined by harmonic potentials [52], or the oscillation of a superconductor ring levitated by a magnetic field [53].…”

Section: Literature Reviewmentioning

confidence: 99%

Effects of a thermal inversion experiment on STEM students learning and application of damped harmonic motion

Peña¹,

Morán-Soto²,

Rodríguez-Masegosa³

et al. 2020

Preprint

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There are diverse teaching methodologies to promote both collaborative and individual work in undergraduate physics courses. However, few educational studies seek to understand how students learn and apply new knowledge through open-ended activities that require mathematical modeling and experimentation focused on environmental problems. In this work, we propose a novel home experiment to simulate the dynamics of a particulate under temperature inversion and model it as damped harmonic motion. Twenty six first year students enrolled in STEM majors answered six qualitative questions after designing and developing the experiment. These questions helped analyze the students epistemological beliefs about their learning process of physics topics and its applications. Results showed that this type of open-ended experiments could facilitate the students understanding of physics phenomena. In addition, this experiment showed that it could help physics professors to promote students epistemological development by giving their students the opportunity to search for different sources of knowledge and becoming self-learners instead of looking at the professor as the epistemological authority. At the end, students described this activity as a positive experience that helped them realize alternative ways to apply physics topics in different contexts of their environment.

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On Noether's theorem for the invariant of the time-dependent harmonic oscillator

Abstract: The time-dependent oscillator describing parametric oscillation, the concept of invariant and Noether's theorem are important issues in physics education. Here, it is shown how they can be interconnected in a simple and unified manner.

Cited by 6 publications

References 17 publications

Weak invariants in dissipative systems: action principle and Noether charge for kinetic theory

Weak invariants in dissipative systems: action principle and Noether charge for kinetic theory

Effects of a Thermal Inversion Experiment on STEM Students Learning and Application of Damped Harmonic Motion

Effects of a thermal inversion experiment on STEM students learning and application of damped harmonic motion

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