Euler′s thought on the invariance of the fundamental equation of the calculus of variations and its origin

Aim To explore Euler′s thought on the invariance of the fundamental equation of the calculus of variations and its background.Methods Historic analysis and literatural review.Results Euler′s thought on the invariance of the fundamental equation of the calculus of variations was closely connected with the change of the subject of analysis in the 18th century.With the separation of analysis subject from geometry,the formula involving letters and numbers or the functions as analytical expressions gradually replaced the geometric curves and eventually became the fundamental concept of analysis subject.The thought of Euler reflected the change at that time.Conclusion Euler′s thought of the invariance of the fundamental equation was the result of the change of the subject of analysis in the 18th century.Following this congnitive guidance,Lagrange created his formal method of variation based on pure analysis and further expanded the employments of the calculus of variations into dynamical problems.