[The Helmholtz-Cartan theorem for a simple integral of higher order]
(1979) Academie des Sciences. Comptes Rendus Hebdomadaires des Seances. Serie A. Sciences Mathematiques — Vol. 288, n° 17, p. 827-830 (1979)
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- Abstract
- The equivalence between the principles of classical mechanics and the theorem which Elie Cartan called principle of conservation of impulse-energy originates in the possibility of replacing Hamilton's principle, delta I=0 where I is the integral of L(t,q/sup i/,dq /sup i//dt)dt, 1<or=i<or=n, (first order variational problem P /sub 1/ on (t,q/sup i/)-space) by a stronger and zero-order variational principle P /sub 0/, Delta J=0 where J is the integral of the 1-form omega =Ltd+ Sigma ( part L/ part q/sup i /))dq/sup i/-dq/dt/sup i/dt)= Sigma p/sub i/dq/sup i /-Hdt of the phase space (t,q/sup i/,dq/dt/sup i/). This possibility is extended here, and consequently all the formalism of Hamilton-Jacobi-E. Cartan, to order s>or=1 variational problems, the Lagrangians L of which depend on the derivatives of d/sup k/q/sup i //dt/sup k/ of order k<or=s.
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Dedecker, P. (1979). [The Helmholtz-Cartan theorem for a simple integral of higher order]. Academie des Sciences. Comptes Rendus Hebdomadaires des Seances. Serie A. Sciences Mathematiques, 288(17), 827-830. https://hdl.handle.net/2078.5/149124 (Original work published 1979)