# 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

## 1 Answer

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.

Hamilton's formulation is not directly variational, although in hindsight it can be interpreted in terms of optimal control theory developed in 1950s. Being equivalent one can always be transformed into the other of course, but Hamilton did discover the formulation attributed to him in 1834. See The Early History of Hamilton- Jacobi Dynamics 1834-1837.