15.7: Symplectic Representation
- Page ID
- 14242
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Hamilton’s first-order equations of motion are symmetric if the generalized and constraint force terms, in equation \((15.1.9)\), are excluded.
\[\mathbf{\dot{q}} = \frac{\partial H}{ \partial \mathbf{p}} \quad − \mathbf{\dot{p}} = \frac{\partial H}{ \partial \mathbf{q}} \nonumber\]
This stimulated attempts to treat the canonical variables \((\mathbf{q}, \mathbf{p})\) in a symmetric form using group theory. Some graduate textbooks in classical mechanics have adopted use of symplectic symmetry in order to unify the presentation of Hamiltonian mechanics. For a system of \(n\) degrees of freedom, a column matrix \(\boldsymbol{\eta}\) is constructed that has \(2n\) elements where
\[\eta_j = q_j \quad \eta_{n+j} = p_j \quad j \leq n \label{15.150}\]
Therefore the column matrix
\[\left(\frac{\partial H}{ \partial \boldsymbol{\eta}} \right)_j = \frac{\partial H }{\partial q_j} \quad \left(\frac{\partial H}{ \partial \boldsymbol{\eta}} \right)_{n+j} = \frac{\partial H }{\partial p_j} \quad j \leq n \label{15.151}\]
The symplectic matrix \(\mathbf{J}\) is defined as being a \(2n\) by \(2n\) skew-symmetric, orthogonal matrix that is broken into four \(n \times n\) null or unit matrices according to the scheme
\[\mathbf{J} = \begin{pmatrix} [\mathbf{0}] & +[\mathbf{1}] \\ − [\mathbf{1}] & [\mathbf{0}] \end{pmatrix} \label{15.152}\]
where \([\mathbf{0}]\) is the \(n\)-dimension null matrix, for which all elements are zero. Also \([\mathbf{1}]\) is the \(n\)-dimensional unit matrix, for which the diagonal matrix elements are unity and all off-diagonal matrix elements are zero. The \(\mathbf{J}\) matrix accounts for the opposite signs used in the equations for \(\mathbf{\dot{q}}\) and \(\mathbf{\dot{p}}\). The symplectic representation allows the Hamilton’s equations of motion to be written in the compact form
\[\boldsymbol{\dot{\eta}} = \mathbf{J}\frac{\partial H }{\partial \boldsymbol{\eta}} \label{15.153}\]
This textbook does not use the elegant symplectic representation since this representation ignores the important generalized forces and Lagrange multiplier forces.