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National Bird Day: Why Birds Matter to Us

National Bird Day is celebrated every year on 5 January . This day is observed to spread awareness about birds, their importance in nature, and the need to protect them. Birds are beautiful living beings and play a very important role in keeping our environment healthy. Origin of National Bird Day National Bird Day was first celebrated in the year 2002 . It was started by bird lovers and environmental groups to protect birds from dangers like deforestation, pollution, and illegal hunting. The main aim of this day is to teach people, especially students, why birds are important and how we can help save them. Why Is National Bird Day Celebrated Every Year? National Bird Day is celebrated every year because many bird species are disappearing due to human activities. Cutting trees, using plastic, pollution, and climate change are harming birds and their homes. This day reminds us that: Birds need protection Nature should be respected Everyone has a responsibil...
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Invariance of Poisson bracket under canonical transformation | Classical Mechanics

Invariance of Poisson bracket under canonical transformation

  • Let u and v be two functions such that u = u (qi, pi, t) and v = v (qi, pi, t)
  • Let a canonical transformation is from (qi, pi, t) → (Qi, Pi, t)
  • Here q = q (Q, P, t) and p = p (Q, P, t)
  • Corresponding to it the transformation in u and v are u (qi, pi, t) → u′ (Qi, Pi, t) and v (qi, pi, t) → v′ (Qi, Pi, t)
  • Now we have to prove that if (q, p, t) → (Q, P, t) is canonical then [u, v]p, q = [u′, v′]P, Q
  • It means the Poisson bracket are invariant under a canonical transformation.

Proof

  • If F1 and F2 are generating function, then the transformation relation for the variables are F2 = F1 + PQ

  • Thus the Poisson brackets are invariant under a canonical transformation.

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