Lagrange and Hamilton Integrals and their Connection

Abstract

Abstract The object of the following is the change from Lagrange line integrals of single particles to Lagrange line integrals of systems of particles under the condition of the first law. The first problem is the historical variational principle as imagined by Bernoulli, whereas the second problem concerns the built up of thermodynamic systems from elementary integrals. Since thermodynamics disregards mechanisms governing the connection or the interaction of particles the built up of thermodynamic systems is possible if Lagrange integrals exhibiting addition theorems between the limits are taken into account. Hilbert’s Independence Theorem involving the transition from the Lagrange to the Hamilton potential proved to be a suitable method to turn over from the mathematical Lagrange potentials to the Hamilton representation as used in physics. Caratheodory, basing his “Complete Figures of Calculus of Variation” on Hilberts conception has given the variational principle that minimizes the connection of the elementary Lagrange integrals being connected between the limits. The extension of thermodynamics to integrals composed from components showing a connection between the limits yields important thermodynamic properties.