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

:$$

[ 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 $$Q=Q(q,p),\; P=P(q,p)$$ is a canonical transformation, then the Lagrange bracket is an invariant of the transformation, in the sense that $$[\; u,\; v]\_\{q,p\}\; =\; [u\; ,\; v]\_\{Q,P\}$$ 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 $$\backslash Omega\; =\; \backslash frac\; 12\; \backslash Omega\_\{ij\}\; du^i\; \backslash wedge\; du^j$$ where the matrix $$\backslash Omega\_\{ij\}\; =\; [\; u\_i,\; u\_j\; ]\_\{p,q\},\; \backslash quad\; 1\backslash leq\; i,j\backslash leq\; 2n$$ represents the components of {{math|Ω}}, viewed as a tensor, in the coordinates u. This matrix is the inverse of the matrix formed by the Poisson brackets $$\backslash left(\backslash Omega^\{-1\}\backslash right)\_\{ij\}\; =\; \backslash \{u\_i,\; u\_j\backslash \},\; \backslash quad\; 1\; \backslash leq\; i,j\; \backslash leq\; 2n$$ 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 $$[Q\_i,\; Q\_j]\_\{p,q\}=0,\; \backslash quad\; [P\_i,P\_j]\_\{p,q\}=0,\backslash quad\; [Q\_i,\; P\_j]\_\{p,q\}=-[P\_j,\; Q\_i]\_\{p,q\}=\backslash delta\_\{ij\}.$$

{{Main|Canonical transformation}}

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

Consider the following canonical transformation:$$\backslash eta\; =$$

\begin{bmatrix}

q_1\\

\vdots \\

q_N\\

p_1\\

\vdots\\

p_N\\

\end{bmatrix} \quad \rightarrow \quad \varepsilon =

\begin{bmatrix}

Q_1\\

\vdots \\

Q_N\\

P_1\\

\vdots\\

P_N\\

\end{bmatrix}

Defining $M\; :=\; \backslash frac\{\backslash partial\; (\backslash mathbf\{Q\},\; \backslash mathbf\{P\})\}\{\backslash partial\; (\backslash mathbf\{q\},\; \backslash mathbf\{p\})\}$, the Lagrange matrix is defined as $\backslash mathcal\; L(\backslash eta)\; =\; M^TJM$

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

$$\backslash mathcal\; L\_\{ij\}(\backslash eta)\; =\; [M^TJM]\_\{ij\}\; =\; \backslash sum\_\{k=1\}^\{N\}\; \backslash left(\backslash frac\{\backslash partial\; \backslash varepsilon\_k\}\{\backslash partial\; \backslash eta\_\{i\}\}\; \backslash frac\{\backslash partial\; \backslash varepsilon\_\{N+k\}\}\{\backslash partial\; \backslash eta\_j\}\; -\; \backslash frac\{\backslash partial\; \backslash varepsilon\_\{N+k\}\}\{\backslash partial\; \backslash eta\_i\; \}\; \backslash frac\{\backslash partial\; \backslash varepsilon\_\{k\}\}\{\backslash partial\; \backslash eta\_j\}\backslash right)\; =\; \backslash sum\_\{k=1\}^\{N\}\; \backslash left(\backslash frac\{\backslash partial\; Q\_k\}\{\backslash partial\; \backslash eta\_\{i\}\}\; \backslash frac\{\backslash partial\; P\_\{k\}\}\{\backslash partial\; \backslash eta\_j\}\; -\; \backslash frac\{\backslash partial\; P\_\{k\}\}\{\backslash partial\; \backslash eta\_i\; \}\; \backslash frac\{\backslash partial\; Q\_\{k\}\}\{\backslash partial\; \backslash eta\_j\}\backslash right)=\; [\backslash eta\_i,\backslash eta\_j]\_\backslash varepsilon$$

The Lagrange matrix satisfies the following known properties:$$\backslash begin\{align\}$$

\mathcal L^T &= - \mathcal L \\

|\mathcal L| &= {|M|^2}\\

\mathcal L^{-1}(\eta)&= -M^{-1} J (M^{-1})^T = - \mathcal P(\eta)\\

\end{align}

where the $\backslash mathcal\; P(\backslash eta)$

is known as a Poisson matrix and whose elements correspond to Poisson brackets. The last identity can also be stated as the following:$$\backslash sum\_\{k=1\}^\{2N\}\; \backslash \{\backslash eta\_i,\backslash eta\_k\backslash \}[\backslash eta\_k,\backslash eta\_j]\; =\; -\backslash delta\_\{ij\}$$

Note that the summation here involves generalized coordinates as well as generalized momentum.

The invariance of Lagrange bracket can be expressed as: $[\backslash eta\_i,\backslash eta\_j]\_\backslash varepsilon=[\; \backslash eta\_i,\backslash eta\_j]\_\backslash eta\; =\; J\_\{ij\}$

, which directly leads to the symplectic condition: $M^TJM\; =\; J$

.{{Cite book |last=Giacaglia |first=Giorgio E. O. |title=Perturbation methods in non-linear systems |date=1972 |publisher=Springer |isbn=978-3-540-90054-2 |series=Applied mathematical sciences |location=New York Heidelberg |pages=8–9}}

• Lagrangian mechanics
• Hamiltonian mechanics

{{Reflist}}

• 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. {{MathSciNet|id=1659212}}

• {{mathworld |urlname=LagrangeBracket |title=Lagrange bracket|author=Eric W. Weisstein}}
• {{SpringerEOM|author=A.P. Soldatov|title=Lagrange bracket}}

Category:Bilinear maps

Category:Hamiltonian mechanics