# The equvalence of the virtual work and the Hamiltonian equations

I am reading Whittaker's Analytical Dynamics. This is chapter 10 *Hamiltonian Systems&. Paragraph 109 is Hamiltonian Systems & Their integral invariants.

Whittaker starts with the Lagrangian form of a conservative holonomial system and deduces the Hamiltonian form for the equation of the virtual work

$$\delta H=\sum_{r=1}^n \dot q_r \delta p_r-\dot p_r\delta q_r$$

which he says is equivalent (with no further explanation) to the system

$$\frac{dq_r}{dt}=\frac{\partial H}{\partial p_r},\quad \frac{dp_r}{dt}=-\frac{\partial H}{\partial q_r},\quad (r=1.\dots,n)$$

where $H$ is the Hamiltonian, $q_i$ are the coordinates and $p_i=\frac{\partial L}{\partial\dot q_i}$. $L$ is the Lagrangian.

How does one deduce this equivalence? Furthermore, how does one deduce the symmetric form of the equation of the virtual work $$\delta (\sum_{r=1}^n p_rdq_r-Hdt)=d(\sum_{r=1}^n p_r\delta q_r-H\delta t)$$

• I added two tags. – Carl Mummert Jan 6 '14 at 1:22