Poisson brackets | Identities of Poisson brackets | Classical Mechanics

Poisson brackets and its identities

Poisson brackets

• A Poisson bracket is a special kind of relation between a pair of dynamical variables of any holonomic system, which is found to remain invariant under any canonical transformation.
• They are used to construct new integrals of motion from the known integrals.
• They are classical analogues of commutation relation between operators in quantum mechanics.
• If u (p, q, t) and v (p, q, t) are two dynamical variables, then the Poisson bracket of these quantities with respect to canonical variables (p, q) is

                

Identities of Poisson brackets

• [u, v] = – [v, u]

• Thus the Poisson bracket of any two dynamical variables is anti-commutative.
• If u = v, then

                

• [u, u]_{(p, q)} = 0
• [u, u] = [v, v] = 0

• If c is any constant, then [cu, v] = [u, cv] = c [u, v]

• Similarly [u, cv] = c [u, v]
• ∴  [cu, v] = [u, cv] = c [u, v]

• The Poisson brackets satisfy the distributive property
• [u + v, w] = [u, w] + [v, w] and [u, v w] = [u, v]w + v[u, w]

• Similarly [u, v w] = [u, v]w + v[u, w]

• The partial derivative of Poisson bracket is

• Jacobi identity of Poisson bracket is [u [v, w]] + [v [w, u]] + [w [u, v]] = 0

• If F (w₁, w₂, …, w_n) be a differentiable function of w₁, w₂, …, w_n and all w’s be the function of (p, q, t), then

• Let F (w₁, w₂) be a differentiable function of w₁ and w₂

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