A Revisit to Higher Variations of a Functional
DOI:
https://doi.org/10.53555/nnms.v6i2.523Keywords:
revisit, higher, variations, functionalAbstract
Two definitions of higher variations of a functional can be found in the literature of variational principles or calculus of variations, which differ by only a positive coefficient number. At first glance, such a discrepancy between the two definitions seems to be purely due to a definition-style preference, as when they degenerate to the first variation it leads to the same result. The use of higher (especially second) variations of a functional is for checking the sufficient condition for the functional to be a minimum (or maximum), and both definitions also lead to the same conclusion regarding this aspect. However, a close theoretical study in this paper shows that only one of the two definitions is appropriate and the other is advised to be discarded. A theoretical method is developed to derive the expressions for higher variations of a functional, which is used for the above claim.
References
Bathe, K.J.: Finite element Procedures. Prentice Hall, New Jersey (1996)
Bolza, O.: Lectures on the calculus of variations. University of Chicago Press, Chicago (1904)
Capodanno, P.: Calculation of the second variation in the problem of the stability of the steady motion of a rigid body containing a liquid 61(2), 319–324 (1997)
Carlson, D.A., Leitmann, G.: Fields of extremals and sufficient conditions for the simplest problem of the calculus of variations in n-variables. Springer Optimization and Its Applications 33, 75–89 (2009)
Chen, W.: Variational Principles in Mechanics. Tongji University Press, Shanghai (1987)
Courant, R., D., H.: Methods of Mathematical Physics, Vol. I, 1st edn. Interscience Publishers, Inc., New York (1953)
Fraser, C.G.: Sufficient conditions, fields and the calculus of variations. Historia Mathematica 4(36), 420–427 (2009). DOI 10.1016/j.hm.2009.02.001. URL http://dx.doi.org/10.1016/j.hm.2009.02.001
Gelfand, M., Fomin, S.: Calculus of Variations. Prentice-Hall, Inc., New Jersey (1963)
Goldstine, H.: A History of the Calculus of Variations from the 17th through the 19th Century, 1st edn. Springer-Verlag, New York (1980)
Hu, H.: The variational principle of elasticity and its application. China Science Publishing and Media Ltd., Beijing (1981)
Kot, M.: A First Course in the Calculus of Variations. American Mathematical Society, Rhode Island (2000)
Lanczos, C.: The Variational Principles of Mechanics. University of Toronto Press, Toronto (1952)
Langhaar, H.: Energy Methods in Applied Mechanics. John Wiley and Sons, Inc., New York (1962)
Lao, D.: Fundamentals of the Calculus of Variations, 2nd edn. National Defense Industry Pres, Beijing (2011)
Long, Y., Liu, G., He, F., Luo, X.: New Discussions on Energy theorems. China Architecture and Building Press, Beijing (2004)
Mesterton-Gibbons, M.: A Primer on the Calculus of Variations and Optimal Control Theory. American Mathematical Society, Rhode Island (2000)
Osgood, W.F.: Sufficient Conditions in the Calculus of Variations. Annals of Mathematics 2(1/4), 105–129 (1900)
Simpson, H.C., Spector, S.J.: On the positivity of the second variation in finite elasticity. Archive for Rational Mechanics and Analysis 98(1), 1–30 (1987). DOI10.1007/BF00279960
Suo, X., Combescure, A.: Second variation of energy and an associated line independent integral in fracture mechanics. I-Theory. European Journal of Mechanics - A/Solids 11(5), 609–624 (1992)
Suo, X.Z., Valeta, M.P.: Second variation of energy and an associated line independent integral in fracture mechanics. II. Numerical validations. European Journal of Mechanics, A/Solids 17(4), 541–565 (1998). DOI 10.1016/S0997-7538(99)80022-9
Tauchert, T.: Energy Principles in Structural Mechanics. McGraw-Hill, Inc., New York (1974)
Todhunter, I.: A History of the Progress of the Calculus of Variations During the Nineteenth Century. Macmillan and Co., Cambridge (1861)
Washizu, K.: Variational Methods in Elasticity and Plasticity. Pergamon Press, Oxford (1968)
Weinstock, R.: Calculus of Variations: with Applications to Physics and Engineering, revised edn. Dover Publications, INC., New York (1974)
Zienkiewicz, O.C., Taylor, R.L., Zhu, J.: The Finite Element Method: Its Basis and Fundamentals, 7th edn. (2013)
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Terms & Condition
Submission -
Author can submit the manuscript through our online submission process or email us at the designated email id in contact details.
The other mode of submission not accepted than online and email.
Before submission please read the submission guidelines.
NN Publication accepts only article submitted in pdf/doc/docx/rtf file format. Another format except given file formats will no be considered .
Author will be responsible for the error mistakes in the submission files. The minor changes can be done without any cost after publication. But for major changes NN Publication may charges you the editing charges.
Publication (Online) -
The online publication is scheduled on last date of every month, but it can be delayed by 24 to 48 hours due to editorial process if huge number of articles comes to publish in single issue.
Automatic notificatation email will be sent to the all users on publication of an issue, so its author’s duty to check their email inbox or SPAM folder to get this notification.
After publication of article author can not withdraw their article.
If editor’s found any issue after publication of article then the NN Publication have the authority to remove the article from online website.
No refund will be provided after online publication of article.
Publication (Print) -
The print copy publication are sent as per the author’s request after 2 weeks of online publication of that issue.
NN Publication will ship the article by India Post and provide the consignment number on dispatch of print copy.
NN Publication follows all the guidelines of delivery provided by IndiaPost and hence not responsible for delay in delivery due to any kind of reasons.
Refund of hard copy will not be provided after dispatch or print of the journal.
NN Publication will be responsible for raise a complain if there is any issue occurs in delivery, but still will not be responsible for providing the refund.
NN Publication will be responsible to resend the print copy only and only if the print copy is lost or print copy is damaged in delivery / or there is delay more than 6 months.
According to India Post the delivery should be completed with in 1-3 weeks after dispatch of articles.
Privacy Policy-
NN Publicationl uses the email ids of authors and editors and readers for sending editorial or publication notification only, we do not reveal or sell the email ids to any other website or company.