$$\require{cancel}$$

15.7: Symplectic Representation

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.