Question

Prove the relation between lagranges and poissons bracket.

(ui, uj). {ui, uk) = deljk (kronecker delta function)

Where ( ui, uj) = lagranges bracket

{ui, uk}=poissons bracket

Answer 1

We can show that

Here $\{u_l, u_i\}$ is Lagrange bracket and $[u_l, u_j]$ is Poisson bracket. Now

The first four terms on R.H.S on multiplication is

But

The first four terms on R.H.S on multiplication is

The last four terms will be

The second term is

since

Similarly the third term will be zero.

Hence R.H.S of equation (1) is

Therefore