# What did Lagrange do with his quantity (the Lagrangian in classical mechanics)?

When I was learning classical mechanics, I was quite baffled by the Hamilton's principle, since it involves a quantity named after Lagrange. So, it seems that the principle was not discovered by Lagrange himself. A natural question is then, what did Lagrange do with his quantity?

• Do you mean "not discovered by Hamilton himself", the second sentence is confusing? – Conifold Jul 13 '15 at 21:04
• I mean, Hamilton used the quantity named after Lagrange to do something, so what did Lagrange himself do with his quantity? I would expect the Hamilton's principle to be named as Lagrange's principle. – kaiser Jul 13 '15 at 21:07
• – Peter Taylor Jul 25 '15 at 8:37

Lagrange and Hamilton principles are different ways to formulate the same Newtonian mechanics. One uses positions and momenta as variables, the other positions and velocities. But they both use kinetic $T$ and potential $V$ energies to write down Hamiltonian $\mathcal{H}=T+V$ and Lagragian $L=T-V$ functions respectively, which lead to Hamilton and Euler-Lagrange equations of motion respectively. Lagrange's formulation is variational, he integrates $L$ and takes the variation to obtain the equations, in this he was anticipated by Maupertuis.