Modern Introduction to Classical Mechanics and Control

Burghes, David; Downs, Angela M.; Hadlock, Charles R.

doi:10.1119/1.10840

Cited by 3 publications

(6 citation statements)

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“…In optimal control theory (see e.g. Burghes & Downs 1975; Kirk 2004), the control must maximize the so‐called objective function, J , which is written as where L is the Lagrangian which quantifies the constraints on the system and is a function of the state and control variables as shown. For a minimization, the appropriate objective function is simply the maximizing objective function multiplied by −1.…”

Section: Active Galactic Nucleus Feedback Models With Additional Comentioning

confidence: 99%

“…Optimal control theory also makes use of Pontryagin's maximum principle, which states that the optimal state q *( t ), optimal control α*( t ) and corresponding costate vector y *( t ) must maximize the Hamiltonian (e.g. Burghes & Downs 1975). Therefore, the condition for optimality is For a complete solution, it is also necessary to solve the Euler–Lagrange equations of the Hamiltonian: Full solutions can then be obtained by applying the appropriate boundary conditions, which depend on the specific problem.…”

Section: Active Galactic Nucleus Feedback Models With Additional Comentioning

confidence: 99%

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Investigating a model of optimized active galactic nucleus feedback

Pope¹

2011

Monthly Notices of the Royal Astronomical Society

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The feedback heating from active galactic nuclei (AGNs) in massive galaxies and galaxy clusters can be thought of as a naturally occurring control system which plays a significant role in regulating both star formation rates and the X-ray luminosity of the surrounding hot gas. In the simplest case, negative feedback can be viewed as a system response that is 'optimized' to minimize deviations from equilibrium, such that the system rapidly evolves towards a steady state. However, a general solution of this form appears to be incompatible with radio observations which indicate intermittent AGN outbursts. Here, we explore an energetically favourable scenario in which feedback is required to both balance X-ray gas cooling and minimize the sum of the energy radiated by the gas and the energy injected by the AGN. This specification is equivalent to ensuring that AGN heating balances the X-ray gas cooling with the minimum black hole growth. It is shown that minimum energy heating occurs in discrete events and not at a continuous, constant level. Furthermore, systems with stronger feedback experience proportionally more powerful heating events, but correspondingly smaller duty cycles. Interpreting observations from this perspective would imply that stronger feedback occurs in less-massive objects -elliptical galaxies rather than galaxy clusters. One direct consequence of this effect would be that AGN heating events are sufficiently powerful to expel hot gas from the gravitational potential of a galaxy, but not a galaxy cluster, which is consistent with theoretical explanations for the steepening of the L X -T relation at temperatures below 1-2 keV.

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Section: Active Galactic Nucleus Feedback Models With Additional Comentioning

confidence: 99%

Section: Active Galactic Nucleus Feedback Models With Additional Comentioning

confidence: 99%

Investigating a model of optimized active galactic nucleus feedback

Pope¹

2011

Monthly Notices of the Royal Astronomical Society

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“…In optimal control theory (see Burghes & Downs 1975;Kirk 2004, for example), the control must maximise the socalled objective function, J, which is written as…”

Section: Optimised Agn Feedbackmentioning

confidence: 99%

Section: Optimised Agn Feedbackmentioning

confidence: 99%

Investigating a model of optimised AGN feedback

Pope

2011

Preprint

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Feedback heating from AGN in massive galaxies and galaxy clusters can be thought of as a naturally occurring control system which plays a significant role in regulating both star formation rates and the X-ray luminosity of the surrounding hot gas. In the simplest case, negative feedback can be viewed as a system response that is 'optimised' to minimise deviations from equilibrium, such that the system rapidly evolves towards a steady state. However, a general solution of this form appears to be incompatible with radio observations which indicate intermittent AGN outbursts. Here, we explore an energetically favourable scenario in which feedback is required to both balance X-ray gas cooling, and minimise the sum of the energy radiated by the gas and the energy injected by the AGN. This specification is equivalent to ensuring that AGN heating balances the X-ray gas cooling with minimum black hole growth. It is shown that minimum energy heating occurs in discrete events, and not at a continuous, constant level. Furthermore, systems with stronger feedback experience proportionally more powerful heating events, but correspondingly smaller duty cycles. Interpreting observations from this perspective would imply that stronger feedback occurs in less massive objects -elliptical galaxies, rather than galaxy clusters. One direct consequence of this effect would be that AGN heating events are sufficiently powerful to expel hot gas from the gravitational potential of a galaxy, but not a galaxy cluster, which is consistent with theoretical explanations for the steepening of the L X − T relation at temperatures below 1-2 keV.

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“…In order to apply the results of $2 to the construction of Lagrangians for one-dimensional systems with variable mass, we first require the corresponding equation of motion. Many mechanics texts (Goodman and Warner 1964, Burghes and Downs 1975, Griffiths 1985 derive this equation in the following manner. With respect to a chosen inertial frame of reference, the momentum of the body at time t + dt is (m(?)…”

Section: Lagrangians For Simple Systems With Variable Massmentioning

confidence: 99%

Lagrangians for simple systems with variable mass

Leubnert¹,

Krumm²

1990

Eur. J. Phys.

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The Lagrangian formulation of classical mechanics and its applications figure prominently in the educational literature. Yet systems with variable mass are summarily excluded from this formulation and discussed in terms of Newtonian theory only. This omission is neither technically justified nor desirable from a pedagogical point of view, because it might suggest to the student that such systems are beyond the scope of the Lagrangian approach. Therefore, the authors show that the formalism can be readily extended to include those variable mass systems that are treated in the textbooks in a Newtonian fashion, and its application is illustrated by means of three instructive examples.

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Modern Introduction to Classical Mechanics and Control

Cited by 3 publications

References 0 publications

Investigating a model of optimized active galactic nucleus feedback

Investigating a model of optimized active galactic nucleus feedback

Investigating a model of optimised AGN feedback

Lagrangians for simple systems with variable mass

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