Abstract: In terms of a mathematical expression of the quantitative causal principle, this paper gives a unification of Hamilton, Voss, Hlder, Maupertuis-Lagrange varia tional principles of integral style of the second-order Lagrangians, and finds t he intrinsic relations among all the different integral variational principles. It is proved that f0= 0 is just the result satisfying the quantitati ve ca usal principle． The Noether conservation charges of Hamilton, Voss, Hlder, Ma upertuis-Lagrange variational principles are shown up, and the intrinsic relatio ns among the Noether conservation charges of all the integral variational princ iples are discovered. Our investigations make the expressions of the past scrappy numerous variational principles be unified into a relative consistent system o f all the variational principles in terms of the quantitative causal principle, and show that all the variational principles become deductions of the quantitat ive causal principle.

Keywords: 
• variational principle, causal principle, motion equation, symmetry /