Lagrange bracket

<h2 id="definition">Definition</h2>
Suppose that (q1, ..., qn, p1, ..., pn) is a system of <a href="/facts/Canonical_coordinates/9g2SFFRz">canonical coordinates</a> on a <a href="/facts/Phase_space/qPe4Fq57">phase space</a>. 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
          
        
        =
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          (
          
            
              
                
                  ∂
                  
                    q
                    
                      i
                    
                  
                
                
                  ∂
                  u
                
              
            
            
              
                
                  ∂
                  
                    p
                    
                      i
                    
                  
                
                
                  ∂
                  v
                
              
            
            −
            
              
                
                  ∂
                  
                    p
                    
                      i
                    
                  
                
                
                  ∂
                  u
                
              
            
            
              
                
                  ∂
                  
                    q
                    
                      i
                    
                  
                
                
                  ∂
                  v
                
              
            
          
          )
        
        .
      
    
    {\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).}

<h2 id="properties">Properties</h2>
<ul><li>Lagrange brackets do not depend on the system of <a href="/facts/Canonical_coordinates/9g2SFFRz">canonical coordinates</a> (q, p). If (Q,P) = (Q1, ..., Qn, P1, ..., Pn) is another system of canonical coordinates, so that 
 
 
 
 Q
 =
 Q
 (
 q
 ,
 p
 )
 ,
 P
 =
 P
 (
 q
 ,
 p
 )
 
 
 {\displaystyle Q=Q(q,p),P=P(q,p)}
 
 is a <a href="/facts/Canonical_transformation/J8vfkj37">canonical transformation</a>, then the Lagrange bracket is an invariant of the transformation, in the sense that 
 
 
 
 [
 u
 ,
 v
 
 ]
 
 q
 ,
 p
 
 
 =
 [
 u
 ,
 v
 
 ]
 
 Q
 ,
 P
 
 
 
 
 {\displaystyle [u,v]_{q,p}=[u,v]_{Q,P}}
 
 Therefore, the subscripts indicating the canonical coordinates are often omitted.</li>
<li>If Ω is the <a href="/facts/Symplectic_form/o95tWKBc">symplectic form</a> on the 2n-dimensional phase space W and u1,...,u2n form a system of coordinates on W, the symplectic form can be written as 
 
 
 
 Ω
 =
 
 
 1
 2
 
 
 
 Ω
 
 i
 j
 
 
 d
 
 u
 
 i
 
 
 ∧
 d
 
 u
 
 j
 
 
 
 
 {\displaystyle \Omega ={\frac {1}{2}}\Omega _{ij}du^{i}\wedge du^{j}}
 
 where the matrix 
 
 
 
 
 Ω
 
 i
 j
 
 
 =
 [
 
 u
 
 i
 
 
 ,
 
 u
 
 j
 
 
 
 ]
 
 p
 ,
 q
 
 
 ,
 
 1
 ≤
 i
 ,
 j
 ≤
 2
 n
 
 
 {\displaystyle \Omega _{ij}=[u_{i},u_{j}]_{p,q},\quad 1\leq i,j\leq 2n}
 
 represents the components of Ω, viewed as a <a href="/facts/Tensor/pNjFo5tV">tensor</a>, in the coordinates u. This matrix is the <a href="/facts/Inverse_matrix/fPqXk3V8">inverse</a> of the matrix formed by the Poisson brackets 
 
 
 
 
 
 (
 
 Ω
 
 −
 1
 
 
 )
 
 
 i
 j
 
 
 =
 {
 
 u
 
 i
 
 
 ,
 
 u
 
 j
 
 
 }
 ,
 
 1
 ≤
 i
 ,
 j
 ≤
 2
 n
 
 
 {\displaystyle \left(\Omega ^{-1}\right)_{ij}=\{u_{i},u_{j}\},\quad 1\leq i,j\leq 2n}
 
 of the coordinates u.</li>
<li>As a corollary of the preceding properties, coordinates (Q1, ..., Qn, P1, ..., Pn) 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
 ,
 
 [
 
 P
 
 i
 
 
 ,
 
 P
 
 j
 
 
 
 ]
 
 p
 ,
 q
 
 
 =
 0
 ,
 
 [
 
 Q
 
 i
 
 
 ,
 
 P
 
 j
 
 
 
 ]
 
 p
 ,
 q
 
 
 =
 −
 [
 
 P
 
 j
 
 
 ,
 
 Q
 
 i
 
 
 
 ]
 
 p
 ,
 q
 
 
 =
 
 δ
 
 i
 j
 
 
 .
 
 
 {\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}.}
 
</li></ul>
<h2 id="lagrange-matrix-in-canonical-transformations">Lagrange matrix in canonical transformations</h2>
Main article: <a href="/facts/Canonical_transformation/J8vfkj37">Canonical transformation</a>
The concept of Lagrange brackets can be expanded to that of matrices by defining the Lagrange matrix.
Consider the following canonical transformation:
 
 
 
 η
 =
 
 
 [
 
 
 
 
 q
 
 1
 
 
 
 
 
 
 ⋮
 
 
 
 
 
 q
 
 N
 
 
 
 
 
 
 
 p
 
 1
 
 
 
 
 
 
 ⋮
 
 
 
 
 
 p
 
 N
 
 
 
 
 
 ]
 
 
 
 →
 
 ε
 =
 
 
 [
 
 
 
 
 Q
 
 1
 
 
 
 
 
 
 ⋮
 
 
 
 
 
 Q
 
 N
 
 
 
 
 
 
 
 P
 
 1
 
 
 
 
 
 
 ⋮
 
 
 
 
 
 P
 
 N
 
 
 
 
 
 ]
 
 
 
 
 {\displaystyle \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
 :=
 
 
 
 ∂
 (
 
 Q
 
 ,
 
 P
 
 )
 
 
 ∂
 (
 
 q
 
 ,
 
 p
 
 )
 
 
 
 
 
 {\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}
 
, the Lagrange matrix is defined as 
 
 
 
 
 
 L
 
 
 (
 η
 )
 =
 
 M
 
 T
 
 
 J
 M
 
 
 {\textstyle {\mathcal {L}}(\eta )=M^{T}JM}
 
, where 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is the symplectic matrix under the same conventions used to order the set of coordinates. It follows from the definition that:

 
 
 
 
 
 
 L
 
 
 
 i
 j
 
 
 (
 η
 )
 =
 [
 
 M
 
 T
 
 
 J
 M
 
 ]
 
 i
 j
 
 
 =
 
 ∑
 
 k
 =
 1
 
 
 N
 
 
 
 (
 
 
 
 
 ∂
 
 ε
 
 k
 
 
 
 
 ∂
 
 η
 
 i
 
 
 
 
 
 
 
 
 ∂
 
 ε
 
 N
 +
 k
 
 
 
 
 ∂
 
 η
 
 j
 
 
 
 
 
 −
 
 
 
 ∂
 
 ε
 
 N
 +
 k
 
 
 
 
 ∂
 
 η
 
 i
 
 
 
 
 
 
 
 
 ∂
 
 ε
 
 k
 
 
 
 
 ∂
 
 η
 
 j
 
 
 
 
 
 
 )
 
 =
 
 ∑
 
 k
 =
 1
 
 
 N
 
 
 
 (
 
 
 
 
 ∂
 
 Q
 
 k
 
 
 
 
 ∂
 
 η
 
 i
 
 
 
 
 
 
 
 
 ∂
 
 P
 
 k
 
 
 
 
 ∂
 
 η
 
 j
 
 
 
 
 
 −
 
 
 
 ∂
 
 P
 
 k
 
 
 
 
 ∂
 
 η
 
 i
 
 
 
 
 
 
 
 
 ∂
 
 Q
 
 k
 
 
 
 
 ∂
 
 η
 
 j
 
 
 
 
 
 
 )
 
 =
 [
 
 η
 
 i
 
 
 ,
 
 η
 
 j
 
 
 
 ]
 
 ε
 
 
 
 
 {\displaystyle {\mathcal {L}}_{ij}(\eta )=[M^{T}JM]_{ij}=\sum _{k=1}^{N}\left({\frac {\partial \varepsilon _{k}}{\partial \eta _{i}}}{\frac {\partial \varepsilon _{N+k}}{\partial \eta _{j}}}-{\frac {\partial \varepsilon _{N+k}}{\partial \eta _{i}}}{\frac {\partial \varepsilon _{k}}{\partial \eta _{j}}}\right)=\sum _{k=1}^{N}\left({\frac {\partial Q_{k}}{\partial \eta _{i}}}{\frac {\partial P_{k}}{\partial \eta _{j}}}-{\frac {\partial P_{k}}{\partial \eta _{i}}}{\frac {\partial Q_{k}}{\partial \eta _{j}}}\right)=[\eta _{i},\eta _{j}]_{\varepsilon }}

The Lagrange matrix satisfies the following known properties:
 
 
 
 
 
 
 
 
 
 
 L
 
 
 
 T
 
 
 
 
 
 =
 −
 
 
 L
 
 
 
 
 
 
 
 |
 
 
 
 L
 
 
 
 |
 
 
 
 
 =
 
 
 |
 
 M
 
 
 |
 
 
 2
 
 
 
 
 
 
 
 
 
 
 L
 
 
 
 −
 1
 
 
 (
 η
 )
 
 
 
 =
 −
 
 M
 
 −
 1
 
 
 J
 (
 
 M
 
 −
 1
 
 
 
 )
 
 T
 
 
 =
 −
 
 
 P
 
 
 (
 η
 )
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}{\mathcal {L}}^{T}&=-{\mathcal {L}}\\|{\mathcal {L}}|&={|M|^{2}}\\{\mathcal {L}}^{-1}(\eta )&=-M^{-1}J(M^{-1})^{T}=-{\mathcal {P}}(\eta )\\\end{aligned}}}
 
where the 
 
 
 
 
 
 P
 
 
 (
 η
 )
 
 
 {\textstyle {\mathcal {P}}(\eta )}
 
 is known as a Poisson matrix and whose elements correspond to <a href="/facts/Poisson_bracket/7WUH7VXV">Poisson brackets</a>. The last identity can also be stated as the following:
 
 
 
 
 ∑
 
 k
 =
 1
 
 
 2
 N
 
 
 {
 
 η
 
 i
 
 
 ,
 
 η
 
 k
 
 
 }
 [
 
 η
 
 k
 
 
 ,
 
 η
 
 j
 
 
 ]
 =
 −
 
 δ
 
 i
 j
 
 
 
 
 {\displaystyle \sum _{k=1}^{2N}\{\eta _{i},\eta _{k}\}[\eta _{k},\eta _{j}]=-\delta _{ij}}
 
Note that the summation here involves generalized coordinates as well as generalized momentum.
The invariance of Lagrange bracket can be expressed as: 
 
 
 
 [
 
 η
 
 i
 
 
 ,
 
 η
 
 j
 
 
 
 ]
 
 ε
 
 
 =
 [
 
 η
 
 i
 
 
 ,
 
 η
 
 j
 
 
 
 ]
 
 η
 
 
 =
 
 J
 
 i
 j
 
 
 
 
 {\textstyle [\eta _{i},\eta _{j}]_{\varepsilon }=[\eta _{i},\eta _{j}]_{\eta }=J_{ij}}
 
, which directly leads to the symplectic condition: 
 
 
 
 
 M
 
 T
 
 
 J
 M
 =
 J
 
 
 {\textstyle M^{T}JM=J}
 
.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Lagrangian_mechanics/sTxyqWz8">Lagrangian mechanics</a></li>
<li><a href="/facts/Hamiltonian_mechanics/2ReerlNA">Hamiltonian mechanics</a></li></ul>

<ul><li><a href="/facts/Cornelius_Lanczos/FRqrbJIb">Cornelius Lanczos</a>, The Variational Principles of Mechanics, Dover (1986), <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-65067-7.</li>
<li>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. <a href="/facts/MR_(identifier)/uP137L11">MR</a><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1659212">1659212</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Eric W. Weisstein</a>. <a href="https://mathworld.wolfram.com/LagrangeBracket.html">"Lagrange bracket"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li>A.P. Soldatov (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Lagrange_bracket">"Lagrange bracket"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Giacaglia, Giorgio E. O. (1972). Perturbation methods in non-linear systems. Applied mathematical sciences. New York Heidelberg: Springer. pp. 8–9. ISBN 978-3-540-90054-2. <a href="978-3-540-90054-2" target="_blank">978-3-540-90054-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Lagrange bracket open-in-new

Lagrange bracket