Hamiltonian mechanics and astrophysical scaling
| English | 中文 | Pinyin |
|---|---|---|
| canonical momentum/kəˈnɒnɪkl məʊˈmentəm/ | 正则动量 | zhèng zé dòng liàng |
| luminosity/ˌluːmɪˈnɒsɪti/ | 光度 | guāng dù |
A decision before an answer
- Energy methods connect a spring oscillator and an orbit, but their coordinates and physical assumptions differ.
- Your goal: Construct a simple Hamiltonian.
Read the relationship
- The canonical momentum is p=∂L/∂qdot. The Legendre transform H=p qdot−L gives a Hamiltonian when velocities can be expressed using coordinates and momenta.
- Use canonical equations.
For H=p²/(2m)+kq²/2, pdot is:
Differentiate H with respect to q and negate.
Use the defining rule
- 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.
- Apply gravitational and radiative scaling.
Doubling distance changes isotropic flux by factor:
Inverse-square scaling gives one quarter.
Check the conditions
- 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.
- Apply gravitational and radiative scaling.
For the oscillator, qdot=p/m and pdot=−kq, hence qddot=−(k/m)q. If an otherwise identical star is twice as far away, received flux is one quarter, although luminosity is unchanged.
For a blackbody of fixed radius, doubling temperature multiplies luminosity by ____.
Stefan–Boltzmann scaling is T^4.
Apply the task format
- Luminosity 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.
- Apply gravitational and radiative scaling.
A Hamiltonian equals total mechanical energy only under the relevant system assumptions; do not infer this universally from its name.
Which answer fits this case?
Construct a simple Hamiltonian
Observed brightness and intrinsic luminosity are identical quantities.
Flux also depends on distance and attenuation.
Keep the distinctions
- canonical momentum 正则动量 — The derivative of the Lagrangian with respect to a generalised velocity.
- luminosity 光度 — Total emitted power, distinct from flux at an observer.
- Construct a simple Hamiltonian.
- Use canonical equations.
- Apply gravitational and radiative scaling.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.