next up previous
Next: 5 Motion of a Up: Lagrangian and Hamiltonian mechanics Previous: 3 The variational principle

Subsections

4 Hamiltonian mechanics

4.1 Generalized momentum and Hamiltonian

In Hamiltonian mechanics we use generalized momentum in place of velocity as a coordinate. The generalized momentum is defined in terms of the Lagrangian and the coordinates (q,$\dot{q}$) :

 
p = $\displaystyle{\frac{\partial L}{\partial\dot q}}$. (21)

The Hamiltonian is defined in terms of the Lagrangian as follows:

 
H(p,q) = p$\displaystyle\dot{q}$ - L(q,$\displaystyle\dot{q}$) (22)

where in the above equation $\dot{q}$ is replaced with a function of (p,q) by solving the definining equation 21 with respect to $\dot{q}$ . We note that this task may be rather hard if the dependence of L on $\dot{q}$ is complicated. Fortunately, in most interesting siuations, L is quadratic in $\dot{q}$ (i.e. T . the kinetic energy is a quadratic function of the velocity). Thus, the equation for $\dot{q}$ in terms of (p,q) is linear.

4.2 Equations of motion

The Lagrangian equation of motion 5 becomes a pair of equations known as the Hamiltonian system of equations:
 
$\displaystyle\dot{q}$ = $\displaystyle{\frac{\partial H}{\partial p}}$,   
$\displaystyle\dot{p}$ = - $\displaystyle{\frac{\partial H}{\partial q}}$. (23)
The second equation of this system is easy to explain. It is simply the Lagrangian equation 5.

The second equation can be explained by using duality. For the sake of this argument we need to assume that L(q,$\dot{q}$) is strictly convex as a function of $\dot{q}$ . It is sufficient to assume that the Hessian

$\displaystyle\left[\frac{\partial^2 L}{\partial \dot q_i\;\partial \dot q_j}\right]$ (24)

is positive definite. We note that for Newtonian equations this matrix is I . The function H in this case is the Legendre transform of L , i.e.

H(p,q) = $\displaystyle\inf_{p}^{}$[p$\displaystyle\dot{q}$ - L(p,$\displaystyle\dot{q}$)] (25)

where the infimum is taken over all p $\in$ $\reals^{n}_{}$ . One can show that L is a Legendre transform of H as well, i.e

L(q,$\displaystyle\dot{q}$) = $\displaystyle\inf_{\dot q}^{}$[p$\displaystyle\dot{q}$ - H(p,q)]. (26)

In particular, the minimum is attained for

$\displaystyle\dot{q}$ = $\displaystyle{\frac{\partial H}{\partial p}}$. (27)

This is exactly the second equation of the system 23.


next up previous
Next: 5 Motion of a Up: Lagrangian and Hamiltonian mechanics Previous: 3 The variational principle
Marek Rychlik
9/2/1997