Question

Show that invariance of poisson bracket with respect to canonical transformation $\large\rm {[Q_{k’} P_{i}]_{q.p} = [Q_{k’} P_{i}]_{Q.P} = δ_{kl^{.}}}$​

Answer 1

Here, we find the particular canonical extension of a coordinate transformation for which the Lagrangians transform by composition with the transformation, with no extra total time derivative terms added to the Lagrangian.

Let L be a Lagrangian for a system. Consider the coordinate transformation q = F (t, q′). The velocities transform by

v=∂0F(t,q′)+∂1F(t,q′)v′. (5.1)  

We obtain a Lagrangian L′ in the transformed coordinates by composition of L with the coordinate transformation. We require that L′(t, q′, v′) = L(t, q, v), so:

L′(t,q′,v′)=L(t,F(t,q′),∂0F(t,q′)+∂1F(t,q′)v′). (5.2)  

The momentum conjugate to q′ is

p′ =∂2L′(t,q′,v′)   =∂2L(t,F(t,q′),∂0F(t,q′)+∂1F(t,q′)v′)∂1F(t,q′)   =p∂1F(t,q′), (5.3)

Answer 2

Answer:

$\huge{\underline{\underline{\sf{ANSWER}}}}$

One of the attempts at combining two sets of Hamilton’s equations into one tries to

take q and p as forming a complex quantity. Show directly from Hamilton’s equations

of motion that for a system of one degree of freedom the transformation

Q = q + ip, P = Q

∗

is not canonical if the Hamiltonian is left unaltered. Can you find another set of

coordinates Q0 and P 0

that are related to Q, P by a change of scale only, and that are

canonical?

Soln: A given transformation is canonical if the Hamilton’s equations are satisfied in

the transformed coordinate system. Therefore, let us evaluate ∂H

∂Q and ∂H