Let $\mathbf a$ and $\mathbf b$ be constant vectors, and let $\mathbf r$ and $\mathbf p$ be generalized positions and conjugate momenta. For $Q=\mathbf a\cdot\mathbf p$ and $P=\mathbf b\cdot\mathbf r$ to form a canonical conjugate pair, find $\mathbf a\cdot\mathbf b$ as an integer.
A canonical pair must obey $\{Q,P\}=1$. Here $\partial Q/\partial r_i=0$, $\partial Q/\partial p_i=a_i$, $\partial P/\partial r_i=b_i$ and $\partial P/\partial p_i=0$. Consequently $\{Q,P\}=\sum_i(0-a_ib_i)=-\mathbf a\cdot\mathbf b$. Equating this to 1 gives $\mathbf a\cdot\mathbf b=-1$.
More Canonical transformations: Poisson bracket. Special theory of relativity: Lorentz transformations, relativistic kinematics, mass-energy equivalence. questions
GATE 2024 PH Q11– If F 1(Q,q)=Qq generates a canonical transformation from (p,q) to (P,Q), which relation is…
GATE 2024 PH Q25– Two spacecraft fly apart along x with individual speeds 0.5c in an inertial frame. The…
GATE 2025 PH Q53– For a one-dimensional Hamiltonian system, consider the canonical transformation…
GATE 2025 PH Q54– The energy of a free relativistic particle of rest mass m moving along x is T. In a…
GATE 2026 PH Q35– A rocket of length 18.0 m is moving at speed 0.9c (where c is the speed of light) parallel…
GATE 2023 PH Q40– A rod PQ of proper length L lies along the x-axis and moves towards the positive x…