Lagrange Bracket

Lagrange brackets are certain expressions closely related to Poisson brackets that were introduced by Joseph Louis Lagrange, but unlike the Poisson brackets, have fallen out of use.

Definition

Suppose that (''q''₁, …, ''q''_''n'', ''p''₁, …, ''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

:
u, v = \sum_^n \left(\frac \frac - \frac \frac \right).

Properties

* Lagrange brackets do not depend on the system of canonical coordinates (''q'', ''p''). If (''Q'',''P'') = (''Q''₁, …, ''Q''_''n'', ''P''₁, …, ''P''_''n'') is another system of canonical coordinates, so that

::$Q=Q(q,p), P=P(q,p)$

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

:: u, v = , v

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

* If ''Ω'' is the symplectic form on the ''2n''-dimensional phase space ''W'' and ''u''_''1'',…,''u''_''2n'' form a system of coordinates on ''W'', the symplectic form can be written as

:: $\Omega = \frac 12 \Omega_ du^i \wedge du^j$

where the matrix

:: \Omega_ = u_i, u_j , \quad 1\leq i,j\leq 2n ::

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

:: $\left(\Omega^\right)_ = \, \quad 1 \leq i,j \leq 2n$

:of the coordinates ''u''.

* As a corollary of the preceding properties, coordinates (''Q''₁, ..., ''Q''_''n'', ''P''₁, …, ''P''_''n'') on a phase space are canonical if and only if the Lagrange brackets between them have the form

:: _i, Q_j=0, \quad _i,P_j=0,\quad _i, P_j=- _j, Q_i=\delta_.

See also

* Lagrangian mechanics
* Hamiltonian mechanics

References

* Cornelius Lanczos, ''The Variational Principles of Mechanics'', Dover (1986), .
* Iglesias, Patrick, ''Les origines du calcul symplectique chez Lagrange'' he origins of symplectic calculus in Lagrange's work L'Enseign. Math. (2) 44 (1998), no. 3-4, 257–277.

External links

*
* {{SpringerEOM, author=A.P. Soldatov, title=Lagrange bracket

Bilinear maps
Hamiltonian mechanics