Lagrange bracket

Suppose that (q₁, …, q_n, p₁, …, p_n) is a system of canonical coordinates on a phase space. If each of them is expressed as a function of two variables, u and v, then the Lagrange bracket of u and v is defined by the formula

${\displaystyle [u,v]_{p,q}=\sum _{i=1}^{n}\left({\frac {\partial q_{i}}{\partial u}}{\frac {\partial p_{i}}{\partial v}}-{\frac {\partial p_{i}}{\partial u}}{\frac {\partial q_{i}}{\partial v}}\right).}$

• Lagrange brackets do not depend on the system of canonical coordinates (q, p). If (Q,P) = (Q₁, …, Q_n, P₁, …, P_n) is another system of canonical coordinates, so that

${\displaystyle Q=Q(q,p),P=P(q,p)}$

is a canonical transformation, then the Lagrange bracket is an invariant of the transformation, in the sense that

${\displaystyle [u,v]_{q,p}=[u,v]_{Q,P}}$

Therefore, the subscripts indicating the canonical coordinates are often omitted.

• If Ω is the symplectic form on the 2n-dimensional phase space W and u₁,…,u_2n form a system of coordinates on W, the symplectic form can be written as

${\displaystyle \Omega ={\frac {1}{2}}\Omega _{ij}du^{i}\wedge du^{j}}$

where the matrix

${\displaystyle \Omega _{ij}=[u_{i},u_{j}]_{p,q},\quad 1\leq i,j\leq 2n}$::

represents the components of Ω, viewed as a tensor, in the coordinates u. This matrix is the inverse of the matrix formed by the Poisson brackets

${\displaystyle \left(\Omega ^{-1}\right)_{ij}=\{u_{i},u_{j}\},\quad 1\leq i,j\leq 2n}$

of the coordinates u.

• As a corollary of the preceding properties, coordinates (Q₁, ..., Q_n, P₁, …, P_n) on a phase space are canonical if and only if the Lagrange brackets between them have the form

${\displaystyle [Q_{i},Q_{j}]_{p,q}=0,\quad [P_{i},P_{j}]_{p,q}=0,\quad [Q_{i},P_{j}]_{p,q}=-[P_{j},Q_{i}]_{p,q}=\delta _{ij}.}$

• Lagrangian mechanics
• Hamiltonian mechanics

• Cornelius Lanczos, The Variational Principles of Mechanics, Dover (1986), ISBN 0-486-65067-7.
• Iglesias, Patrick, Les origines du calcul symplectique chez Lagrange [The origins of symplectic calculus in Lagrange's work], L'Enseign. Math. (2) 44 (1998), no. 3-4, 257–277. MR1659212

• Eric W. Weisstein. "Lagrange bracket". MathWorld.
• A.P. Soldatov (2001) [1994], "Lagrange bracket", Encyclopedia of Mathematics, EMS Press