Classical mechanics
| English | 中文 | Pinyin |
|---|---|---|
| Lagrangian/læˈɡræŋɡɪən/ | 拉格朗日量 | lā gé lǎng rì liàng |
| angular momentum/ˈæŋɡjʊlə məʊˈmentəm/ | 角动量 | jiǎo dòng liàng |
A decision before an answer
- A planet accelerates even when its speed is constant: velocity includes direction.
- Your goal: Apply Newtonian, Lagrangian and Hamiltonian descriptions.
Read the relationship
- Newton’s second law is a vector statement. Choose coordinates and identify constraints before components.
- Analyse rotation, central forces and oscillations.
Doubling spring mass changes angular frequency by:
ω=sqrt(k/m).
Use the defining rule
- Energy conservation applies when work from nonconservative forces is accounted for; momentum conservation needs zero net external impulse.
- Use conservation laws with their conditions.
In an isolated perfectly inelastic collision, conserved quantity is:
Total momentum is conserved; kinetic energy is generally reduced.
Check the conditions
- For small oscillations, linearise around a stable equilibrium. A restoring term proportional to displacement gives harmonic motion.
- Use conservation laws with their conditions.
For a mass m on a spring k, L=(1/2)m xdot²−(1/2)kx². Euler–Lagrange gives m xddot+kx=0. Thus angular frequency is sqrt(k/m). Doubling mass reduces frequency by factor sqrt(2), not two.
For k=16 and m=4, angular frequency is ____ rad/s.
ω=sqrt(16/4)=2.
Apply the task format
- Lagrange’s equations use L=T−V and generalised coordinates. This undergraduate formalism is distinct from school-level force substitution.
- Use conservation laws with their conditions.
Conservation of kinetic energy does not hold in every collision, even when momentum is conserved.
Which answer fits this case?
Apply Newtonian, Lagrangian and Hamiltonian descriptions
Zero torque about a point implies constant angular momentum about it.
dL/dt equals the net torque about the fixed point.
Keep the distinctions
- Lagrangian 拉格朗日量 — Kinetic energy minus potential energy for the standard mechanical system.
- angular momentum 角动量 — The moment of linear momentum about a reference point.
- Apply Newtonian, Lagrangian and Hamiltonian descriptions.
- Analyse rotation, central forces and oscillations.
- Use conservation laws with their conditions.
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.