AH · Hamiltonian mechanics and astrophysical scaling

Energy methods connect a spring oscillator and an orbit, but their coordinates and physical assumptions differ.

Prerequisites: 26, 41.

• Construct a simple Hamiltonian
• Use canonical equations
• Apply gravitational and radiative scaling

The canonical momentum 正则动量 is p=∂L/∂qdot. The Legendre transform H=p qdot−L gives a Hamiltonian when velocities can be expressed using coordinates and momenta.

Hamilton’s equations are qdot=∂H/∂p and pdot=−∂H/∂q. For an ordinary oscillator H=p²/(2m)+kq²/2, these reproduce Newton’s equation.

For a circular gravitational orbit v²=GM/r and period²=4π²r³/(GM). The assumptions include a dominant central mass and a circular approximation; elliptical orbits use the semimajor axis in Kepler’s law. For a circular satellite of mass m, angular momentum magnitude is mrv=m sqrt(GMr), so identical satellites have L proportional to sqrt(r). A radius ratio 9 therefore gives angular-momentum ratio 3, while the period ratio is 27. This distinction follows from the same centripetal-force relation; do not use period scaling for angular momentum.

Luminosity 光度 and received flux obey F=L/(4πd²) for isotropic radiation. A blackbody has L=4πR²σT⁴; its spectral peak shifts inversely with temperature. Distinguish intrinsic luminosity from observed brightness.

For a regular Lagrangian $L=m\dot q^2/2-kq^2/2$, canonical momentum is

Substitute the velocity into the Legendre transform: Combining these gives $\ddot q=-(k/m)q$. The inversion of momentum to velocity is required; it cannot be assumed for every constrained or singular Lagrangian.

A Hamiltonian equals total mechanical energy only under the relevant system assumptions; do not infer this universally from its name.

canonical momentum: The derivative of the Lagrangian with respect to a generalised velocity.

luminosity: Total emitted power, distinct from flux at an observer.