Analytical mechanics

Generalized momentum and the Hamiltonian

Generalized momentum p_i = ∂L/∂q̇_i is conjugate to coordinate q_i; a Legendre transform converts the Lagrangian into the Hamiltonian H.

In Lagrangian mechanics, position and velocity are natural variables. Hamiltonian mechanics replaces velocity with generalized momentum and represents a state by a point in phase space $(q,p)$ . This is especially useful for studying the structure of motion.

p_i=\frac{\partial L}{\partial\dot q_i}, \qquad H(q,p,t)=\sum_i p_i\dot q_i-L(q,\dot q,t)

Definition: Generalized momentum and Hamiltonian

Generalized momentum is conjugate to a coordinate. After the Legendre transform, $H$ is expressed in $(q,p,t)$ . Only for ordinary natural systems in Cartesian coordinates with a velocity-independent potential does $H$ equal the mechanical energy $T+V$ ; this should not be assumed for every system or coordinate choice.

Explore the surface H(q,p)=E of a harmonic oscillator. Its phase-space trajectory stays on a constant-energy curve.

The Legendre transform

If $p_i=\partial L/\partial\dot q_i$ can be inverted to express $\dot q_i$ in terms of $(q,p,t)$ , substitute those velocities into $H=\sum_i p_i\dot q_i-L$ . Local invertibility requires a nonsingular velocity Hessian $\partial^2L/\partial\dot q_i\partial\dot q_j$ . Constrained or singular systems require separate treatment.

L=\frac12m\dot q^2-\frac12m\omega^2q^2 \quad\Longrightarrow\quad H=\frac{p^2}{2m}+\frac12m\omega^2q^2

For the harmonic oscillator, $p=m\dot q$ . Substituting $\dot q=p/m$ gives the Hamiltonian above. The level sets $H=E$ are ellipses in the $(q,p)$ phase plane, and periodic motion runs along an ellipse.

Example: Hamiltonian of a free particle

For $L=\tfrac12m\dot x^2$ , find $p$ and $H$ .

Solution

$p=\partial L/\partial\dot x=m\dot x$ . Hence $H=p\dot x-L=p^2/(2m)$ , the kinetic energy.

Quick check

The Legendre transform is invertible only when the Hessian $W_{ij}=\partial^2L/(\partial\dot q_i\partial\dot q_j)$ is nonsingular. Then velocities can be solved in terms of $(q,p,t)$ and the Hamiltonian is defined on phase space; a singular Hessian instead signals canonical constraints that require separate treatment. A charged particle in electromagnetic potentials is instructive: $L=\tfrac12m\dot{\mathbf r}^{2}+e\mathbf A\cdot\dot{\mathbf r}-e\phi$ gives $\mathbf p=m\dot{\mathbf r}+e\mathbf A$ , not merely the mechanical momentum $m\dot{\mathbf r}$ .

How is generalized momentum defined?

When does the Hamiltonian usually equal $T+V$ ?

References

Herbert Goldstein, Charles Poole, John Safko (2002). Classical Mechanics