고전 역학
도입됨| English | 의미 |
|---|---|
| Lagrangian/læˈɡræŋɡɪən/ | 라그랑지안 |
| angular momentum/ˈæŋɡjʊlə məʊˈmentəm/ | 각운동량 |
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:
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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:
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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.
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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.
다음 중 이 사례에 부합하는 답안은 무엇입니까?
뉴턴, 라그랑주 및 해밀턴 서술법 적용
Zero torque about a point implies constant angular momentum about it.
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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.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.