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UNUSUAL PROPERTIES OF ADIABATIC INVARIANCE IN A BILLIARD MODEL RELATED TO THE ADIABATIC PISTON PROBLEM.

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Title: UNUSUAL PROPERTIES OF ADIABATIC INVARIANCE IN A BILLIARD MODEL RELATED TO THE ADIABATIC PISTON PROBLEM.
Alternate Title: НЕУОБИЧАJЕНЕ ОСОБИНЕ АДИJАБАТСКЕ ИНВАРИJАНТЕ У БИЛИJАРНОМ МОДЕЛУ ПОВЕЗАНОМ СА ПРОБЛЕМОМ АДИJАБАТСКОГ КЛИПА.
Authors: Skinner, Joshua1 skinnerjoshua515@gmail.com, Neishtadt, Anatoly1 a.neishtadt@lboro.ac.uk
Source: Theoretical & Applied Mechanics. 2025, Vol. 52 Issue 1, p9-16. 8p.
Subjects: Conserved quantity, Mathematical invariants, Kinetic energy, Particle motion, Classical mechanics, Elastic scattering
Abstract (English): We consider the motion of two massive particles along a straight line. A lighter particle bounces back and forth between a heavier particle and a stationary wall, with all collisions being ideally elastic. This is one of canonical models in the theory of adiabatic invariants. It is known that if the lighter particle moves much faster than the heavier one, and the kinetic energies of the particles are of the same order, then the product of the speed of the lighter particle and the distance between the heavier particle and the wall is an adiabatic invariant: its value remains approximately constant over a long period. We show that the value of this adiabatic invariant, calculated at the collisions of the lighter particle with the wall, is a constant of motion (i.e., an exact adiabatic invariant). On the other hand, the value of this adiabatic invariant at the collisions between the particles slowly, linearly in time, decays with each collision. The model we consider is a highly simplified version of the classical adiabatic piston problem, where the lighter particle represents a gas particle, and the heavier particle represents the piston. [ABSTRACT FROM AUTHOR]
Abstract (Serbian): Разматрамо кретање две материjалне тачке (честице) дуж праве линиjе. Лакша честица се одбиjа напред-назад између теже честице и непокретног зида, при чему су сви судари идеално еластични. Ово jе jедан од канонских модела у теориjи адиjабатских инвариjанти. Познато jе да ако се лакша честица креће много брже од теже, а кинетичке енергиjе су им истог реда, онда jе производ брзине лакше честице и растоjања између теже честице и зида адиjабатска инвариjанта: њена вредност остаjе приближно константна током дужег периода. Показуjемо да jе вредност ове адиjабатске инвариjанте, израчуната при сударима лакше честице са зидом, константа кретања (тj. тачна адиjабатска инвариjанта). С друге стране, вредност ове адиjабатске инвариjанте при сударима између честица полако, линеарно у времену, опада са сваким сударом. Модел коjи разматрамо jе веома поjедностављена верзиjа класичног адиjабатског проблема клипа, где лакша честица представља честицу гаса, а тежа честица представља клип. [ABSTRACT FROM AUTHOR]
Copyright of Theoretical & Applied Mechanics is the property of Theoretical & Applied Mechanics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
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PubType: Academic Journal
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  Data: UNUSUAL PROPERTIES OF ADIABATIC INVARIANCE IN A BILLIARD MODEL RELATED TO THE ADIABATIC PISTON PROBLEM.
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  Data: НЕУОБИЧАJЕНЕ ОСОБИНЕ АДИJАБАТСКЕ ИНВАРИJАНТЕ У БИЛИJАРНОМ МОДЕЛУ ПОВЕЗАНОМ СА ПРОБЛЕМОМ АДИJАБАТСКОГ КЛИПА.
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  Data: <searchLink fieldCode="AR" term="%22Skinner%2C+Joshua%22">Skinner, Joshua</searchLink><relatesTo>1</relatesTo><i> skinnerjoshua515@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Neishtadt%2C+Anatoly%22">Neishtadt, Anatoly</searchLink><relatesTo>1</relatesTo><i> a.neishtadt@lboro.ac.uk</i>
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  Data: <searchLink fieldCode="JN" term="%22Theoretical+%26+Applied+Mechanics%22">Theoretical & Applied Mechanics</searchLink>. 2025, Vol. 52 Issue 1, p9-16. 8p.
– Name: Subject
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  Data: <searchLink fieldCode="DE" term="%22Conserved+quantity%22">Conserved quantity</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+invariants%22">Mathematical invariants</searchLink><br /><searchLink fieldCode="DE" term="%22Kinetic+energy%22">Kinetic energy</searchLink><br /><searchLink fieldCode="DE" term="%22Particle+motion%22">Particle motion</searchLink><br /><searchLink fieldCode="DE" term="%22Classical+mechanics%22">Classical mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Elastic+scattering%22">Elastic scattering</searchLink>
– Name: Abstract
  Label: Abstract (English)
  Group: Ab
  Data: We consider the motion of two massive particles along a straight line. A lighter particle bounces back and forth between a heavier particle and a stationary wall, with all collisions being ideally elastic. This is one of canonical models in the theory of adiabatic invariants. It is known that if the lighter particle moves much faster than the heavier one, and the kinetic energies of the particles are of the same order, then the product of the speed of the lighter particle and the distance between the heavier particle and the wall is an adiabatic invariant: its value remains approximately constant over a long period. We show that the value of this adiabatic invariant, calculated at the collisions of the lighter particle with the wall, is a constant of motion (i.e., an exact adiabatic invariant). On the other hand, the value of this adiabatic invariant at the collisions between the particles slowly, linearly in time, decays with each collision. The model we consider is a highly simplified version of the classical adiabatic piston problem, where the lighter particle represents a gas particle, and the heavier particle represents the piston. [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label: Abstract (Serbian)
  Group: Ab
  Data: Разматрамо кретање две материjалне тачке (честице) дуж праве линиjе. Лакша честица се одбиjа напред-назад између теже честице и непокретног зида, при чему су сви судари идеално еластични. Ово jе jедан од канонских модела у теориjи адиjабатских инвариjанти. Познато jе да ако се лакша честица креће много брже од теже, а кинетичке енергиjе су им истог реда, онда jе производ брзине лакше честице и растоjања између теже честице и зида адиjабатска инвариjанта: њена вредност остаjе приближно константна током дужег периода. Показуjемо да jе вредност ове адиjабатске инвариjанте, израчуната при сударима лакше честице са зидом, константа кретања (тj. тачна адиjабатска инвариjанта). С друге стране, вредност ове адиjабатске инвариjанте при сударима између честица полако, линеарно у времену, опада са сваким сударом. Модел коjи разматрамо jе веома поjедностављена верзиjа класичног адиjабатског проблема клипа, где лакша честица представља честицу гаса, а тежа честица представља клип. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Theoretical & Applied Mechanics is the property of Theoretical & Applied Mechanics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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    Identifiers:
      – Type: doi
        Value: 10.2298/TAM241121006S
    Languages:
      – Code: eng
        Text: English
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        PageCount: 8
        StartPage: 9
    Subjects:
      – SubjectFull: Conserved quantity
        Type: general
      – SubjectFull: Mathematical invariants
        Type: general
      – SubjectFull: Kinetic energy
        Type: general
      – SubjectFull: Particle motion
        Type: general
      – SubjectFull: Classical mechanics
        Type: general
      – SubjectFull: Elastic scattering
        Type: general
    Titles:
      – TitleFull: UNUSUAL PROPERTIES OF ADIABATIC INVARIANCE IN A BILLIARD MODEL RELATED TO THE ADIABATIC PISTON PROBLEM.
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            NameFull: Skinner, Joshua
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            NameFull: Neishtadt, Anatoly
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              Text: 2025
              Type: published
              Y: 2025
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            – TitleFull: Theoretical & Applied Mechanics
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