Classical mechanics · 经典力学
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| 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.
答案前的判断
- 行星即使速率不变也会加速：速度包含方向信息。
- 你的目标：应用牛顿、拉格朗日和哈密顿描述。
Read the relationship
- Newton’s second law is a vector statement. Choose coordinates and identify constraints before components.
- Analyse rotation, central forces and oscillations.
阅读关系
- 牛顿第二定律是矢量陈述。选择坐标系并识别约束，再进行分量分析。
- 分析旋转、中心力和振动。
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.
运用定义规则
- 当计入非保守力所做的功时适用能量守恒；动量守恒需要净外力冲量为零。
- 结合守恒定律及其适用条件使用。
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.
检查条件
- 对于小振动，在稳定平衡点附近进行线性化。与位移成正比的恢复力项将产生简谐运动。
- 结合守恒定律及其适用条件使用。
对于质量为 m 的物体连接劲度系数为 k 的弹簧，L=(1/2)m xdot²−(1/2)kx²。欧拉-拉格朗日方程给出 m xddot+kx=0。因此角频率为 sqrt(k/m)。质量加倍会使频率降低 sqrt(2) 倍，而非两倍。
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.
应用题目格式
- 拉格朗日方程使用 L=T−V 和广义坐标。这种本科层面的形式体系不同于中学阶段的力代换方法。
- 结合守恒定律及其适用条件使用。
即使动量守恒，动能守恒也不适用于所有碰撞。
Which answer fits this case? · 哪个答案符合此案例？
Apply Newtonian, Lagrangian and Hamiltonian descriptions · 应用牛顿力学、拉格朗日力学和哈密顿力学描述
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.
保持区别
- 拉格朗日量 — 标准力学系统的动能减去势能。
- 角动量 — 相对于参考点的线动量的力矩。
- 应用牛顿力学、拉格朗日力学和哈密顿力学描述。
- 分析旋转、中心力和振动。
- 结合守恒定律及其适用条件使用。
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.