15.7: Symplectic Representation

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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.