Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Introduction Read chapter Read first chapter

2015 | OriginalPaper | Chapter

4. Properties of Minimum Action Curves

Author : Matthias Heymann

Published in: Minimum Action Curves in Degenerate Finsler Metrics

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this chapter we study the properties of minimum action curves, often focusing on a specific subclass of actions. First we show which points minimizing curves can pass “in infinite length.” Then we find for a certain type of Hamiltonian actions that the action of the drift vector field’s flowlines vanishes, and that bending curves into the direction of the drift reduces their action. As a consequence, we then prove the non-existence of minimizers in some situations, and we show that minimizers leading from one attractor of the drift to another have to pass a saddle point on the separatrix between the two basins of attraction.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Existence of Minimum Action Curves
next chapter Conclusions
Literature
1.
Arnold, V.I.: Ordinary Differential Equations, 1st edn. Universitext. Springer, Berlin/Heidelberg/New York (1992)
2.
Bao, D.D.W., Chern, S.S., Shen, Z.: An Introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics, vol. 200. Springer, New York (2000)
3.
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. International Series in Pure and Applied Mathematics. McGraw-Hill, New York/Toronto/London (1955)MATH
4.
5.
Day, M.V., Darden, T.A.: Some regularity results on the Ventcel-Freidlin quasi-potential function. Appl. Math. Optim. 13(1), 259–282 (1985)CrossRefMathSciNetMATH
6.
E, W., Ren, W., Vanden-Eijnden, E.: String method for the study of rare events. Phys. Rev. B 66, 052301 (2002)
7.
E, W., Ren, W., Vanden-Eijnden, E.: Minimum action method for the study of rare events. Commun. Pure Appl. Math. 57(5), 637–656 (2004)
8.
Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Grundlehren der Mathematischen Wissenschaften, vol. 260, 2nd edn. Springer, New York (1998)
9.
Heymann, M.: The geometric minimum action method: A least action principle on the space of curves. Ph.D. thesis, New York University (2007)
10.
Heymann, M., Vanden-Eijnden, E.: The geometric minimum action method: A least action principle on the space of curves. Commun. Pure Appl. Math. 61(8), 1052–1117 (2008)CrossRefMathSciNet
11.
Heymann, M., Vanden-Eijnden, E.: Pathways of maximum likelihood for rare events in nonequilibrium systems: Application to nucleation in the presence of shear. Phys. Rev. Lett. 100, 140601 (2008)CrossRef
12.
13.
14.
Perko, L.: Differential Equations and Dynamical Systems. Texts in Applied Mathematics, vol. 7, 3rd edn. Springer, New York (2001)
15.
Roma, D.M., O’Flanagan, R.A., Ruckenstein, A.E., Sengupta, A.M., Mukhopadhyay, R.: Optimal path to epigenetic switching. Phys. Rev. E 71, 011902 (2005)CrossRef
16.
Shwartz, A., Weiss, A.: Large Deviations for Performance Analysis: Queues, Communications, and Computing. Stochastic Modeling Series. Chapman & Hall, London/Glasgow (1995)MATH
17.
Simon, B.: Semiclassical analysis of low lying eigenvalues, II. Tunneling. Ann. Math., 2nd Ser. 120(1), 89–118 (1984)
18.
Stein, E.M., Shakarchi, R.: Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis. Princeton University Press, Princeton/Oxford (2005)
19.
Vanden-Eijnden, E., Heymann, M.: The geometric minimum action method for computing minimum energy paths. J. Chem. Phys. 128(6), 061103 (2008)CrossRef
20.
Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems, 2nd edn. Springer, Berlin/Heidelberg/New York (2000)
Metadata
Title
Properties of Minimum Action Curves
Author
Matthias Heymann
Copyright Year
2015
Book
Minimum Action Curves in Degenerate Finsler Metrics

Print ISBN: 978-3-319-17752-6

Electronic ISBN: 978-3-319-17753-3

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-17753-3_4