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Wikipedia / Lagrange bracket

Lagrange brackets are certain expressions closely related to Poisson brackets that were introduced by Joseph Louis Lagrange from 1808 to 1810 for the purposes of mathematical formulation of classical mechanics, but unlike the Poisson brackets, have fallen out of use.

Definition

Suppose that (q1, ..., qn, p1, ..., pn) 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

Properties

  • Lagrange brackets do not depend on the system of canonical coordinates (q, p). If (Q,P)=(Q1, ..., Qn, P1, ..., Pn is a canonical transformation Therefore, the subscripts indicating the canonical coordinates are often omitted.

  • If is the symplectic form on the 2n-dimensional phase space W and u1,...,u2n form a system of coordinates on W represents the components of , viewed as a tensor, in the coordinates u. This matrix is the inverse of the coordinates u.

  • As a corollary of the preceding properties, coordinates (Q1, ..., Qn, P1, ..., Pn

Lagrange matrix in canonical transformations

The concept of Lagrange brackets can be expanded to that of matrices by defining the Lagrange matrix.

is the symplectic matrix under the same conventions used to order the set of coordinates. It follows from the definition that:

is known as a Poisson matrix and whose elements correspond to Poisson bracketsNote that the summation here involves generalized coordinates as well as generalized momentum.

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See also

References

  • Cornelius Lanczos, The Variational Principles of Mechanics, Dover (1986), .

  • 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.