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GRE Subject Test · GRE Physics หัวข้อ 26 6:16 การบรรยายภาษาอังกฤษ · คำบรรยายภาษาอังกฤษ + 中文 ลอยตัวบนภาพ
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Transcript
A pendulum's string tension changes all through the swing, yet its equation of motion can be found without ever solving for that tension.
单摆绳子的张力在整个摆动过程中不断变化，但无需求出张力就能得到它的运动方程。
This topic builds the Lagrangian in independent coordinates, uses cyclic coordinates to find conserved momenta, and linearises about a stable equilibrium.
本主题用独立坐标建立拉格朗日量，利用循环坐标找到守恒的广义动量，并在稳定平衡点附近线性化。
A constraint removes an independent coordinate before you form the kinetic energy.
约束在构建动能之前会消除一个独立坐标。
For a fixed-length pendulum choose theta from the downward vertical, so x equals l sine theta and y equals minus l cosine theta.
对于固定长度的单摆，从竖直向下方向选取 θ：x 等于 l sin θ，y 等于负 l cos θ。
Differentiating both coordinates gives v squared equals l squared theta dot squared, not l squared sine squared theta times theta dot squared.
对两个坐标求导可得 v 平方等于 l 平方 θ 点平方，而非 l 平方 sin 平方 θ 乘以 θ 点平方。
Thus T equals m l squared theta dot squared over 2, and V equals m g l times 1 minus cosine theta, with zero potential at the bottom.
因此 T 等于 m l 平方 θ 点平方除以 2，V 等于 m g l 乘以 1 减 cos θ，且以底部为零势能点。
The fixed length is a holonomic constraint; it is a relation among coordinates and possibly time.
固定长度属于完整约束：它是坐标之间以及可能包含时间的关系式。
A rolling velocity constraint needs its own analysis; do not automatically treat every constraint as a coordinate substitution.
滚动速度约束需要单独分析；切勿自动将所有约束都视为坐标代换。
For ideal constraints and the usual conservative system, the stationary-action equation is d by d t of the partial of L with respect to q dot, minus the partial of L with respect to q, equals zero, with L equal to T minus V.
对于理想约束和通常的保守系统，作用量驻值方程为：L 对 q 点的偏导数再对时间求导，减去 L 对 q 的偏导数，等于零，其中 L 等于 T 减 V。
The variations vanish at the two time endpoints.
变分在两个时间端点处为零。
A pendulum gives a partial of L with respect to theta dot of m l squared theta dot, and a partial with respect to theta of minus m g l sine theta, so m l squared theta double dot plus m g l sine theta equals zero.
对于单摆，L 对 θ 点的偏导数为 m l 平方 θ 点，L 对 θ 的偏导数为负 m g l sin θ，因此 m l 平方 θ 双点加 m g l sin θ 等于零。
The momentum derivative acts on every time-dependent factor; for a varying radius, the time derivative of m r squared theta dot contains 2 m r r dot theta dot.
动量项的时间导数作用于每个随时间变化的因子：半径变化时，m r 平方 θ 点对时间的导数包含 2 m r r 点 θ 点。
The variational method does not mean mechanical energy is conserved in a time-dependent system.
变分法并不意味着含时系统的机械能守恒。
For planar central motion, L equals m times r dot squared plus r squared theta dot squared, over 2, minus V of r.
对于平面中心力运动，L 等于 m 乘以 r 点平方加 r 平方 θ 点平方，再除以 2，减去 V(r)。
The angle theta is cyclic because L has no explicit theta dependence, even though L depends on theta dot.
角度 θ 是循环坐标，因为 L 不显含 θ，尽管 L 依赖于 θ 点。
Its canonical momentum, p theta equals m r squared theta dot, is constant.
其正则动量 p θ 等于 m r 平方 θ 点，保持不变。
The radial equation is m r double dot equals m r theta dot squared minus d V by d r.
径向方程为 m r 双点等于 m r θ 点平方减去 dV/dr。
The first term is part of the coordinate acceleration, not a new outward real force in an inertial frame.
第一项是坐标加速度的一部分，而不是惯性系中新出现的真实向外力。
Conservation of p theta makes angular speed increase as r decreases; constant angular speed is a different, externally driven situation.
p θ 守恒使角速度随 r 减小而增大；角速度恒定是另一种由外部驱动的情形。
Near a stable equilibrium q nought, expand V to second order and use a constant local inertia M; V is about V at q nought, plus V double prime at q nought times q minus q nought squared, over 2.
在稳定平衡点 q0 附近，把 V 展开到二阶，并取常数局部惯量 M：V 约等于 V(q0) 加上 V 的二阶导数乘以 q 减 q0 的平方，再除以 2。
Then the small-displacement angular frequency is the square root of V double prime over M.
于是小位移角频率等于 V 的二阶导数除以 M 的平方根。
For a pendulum, sine theta is about theta in radians, so omega equals the square root of g over l; the approximation requires small amplitude.
对于单摆，以弧度计时 sin θ 约等于 θ，因此 ω 等于 g 除以 l 的平方根；该近似要求振幅很小。
A negative V double prime gives instability, not an oscillation with an imaginary measurable frequency.
V 的二阶导数为负表示不稳定，而不是具有可测虚频率的振动。
Check the full nonlinear equation first, then state the approximation and compare units; V double prime over M must have units of inverse time squared.
先检查完整的非线性方程，再说明近似并核对单位：V 的二阶导数除以 M 的单位必须是时间平方的倒数。
With T equal to m l squared theta dot squared over 2 and V equal to m g l times one minus cos theta, the Euler–Lagrange equation reads: the time derivative of m l squared theta dot, plus m g l sine theta, equals zero.
动能 T 等于 m l 平方乘 θ 点平方的一半，势能 V 等于 m g l 乘一减 cos θ，欧拉–拉格朗日方程为：m l 平方 θ 点对时间的导数加 m g l sin θ 等于零。
Divide by m l squared: theta double dot plus g over l sine theta equals zero.
除以 m l 平方：θ 双点加 g 除以 l 乘 sin θ 等于零。
For small angles in radians, sine theta is about theta, so the angular frequency is the square root of g over l.
对于以弧度计的小角度，sin θ 约等于 θ，所以角频率等于 g 除以 l 的平方根。
A bead of mass m slides without friction on a horizontal radial slot rotating at prescribed constant angular speed Omega.
质量为 m 的珠子在水平径向槽内无摩擦滑动，该槽以给定的恒定角速度 Omega 旋转。
Use radius r as coordinate to derive its equation and conserved energy-like quantity.
请以半径 r 为坐标推导其运动方程及守恒的能量型量。
Is its ordinary mechanical energy conserved?
其普通机械能是否守恒？
Explain.
请解释。
Prescribed rotation gives L equals m times dot r squared plus Omega squared r squared, all over 2.
规定旋转给出 L = m(ṙ² + Ω²r²)/2。
Euler-Lagrange gives double dot r equals Omega squared r.
欧拉-拉格朗日方程给出 r̈ = Ω²r。
The time-independent reduced L has conserved dot r times partial derivative of L with respect to dot r minus L, which equals m times dot r squared minus Omega squared r squared, all over 2.
时间无关的约化 L 具有守恒量 ṙ·∂L/∂ṙ − L = m(ṙ² − Ω²r²)/2。
Mechanical energy is m times dot r squared plus Omega squared r squared, all over 2, and is not generally conserved. The rotation drive supplies work through the constraint.
机械能为 m(ṙ² + Ω²r²)/2，通常不守恒：旋转驱动通过约束做功。
Award L, equation, conserved quantity, mechanical-energy distinction and drive.
评分点包括 L、方程、守恒量、机械能区别及驱动。
The bead's angular momentum is not conserved.
珠子的角动量不守恒。
A cyclic coordinate is absent from L itself; its velocity need not be absent.
循环坐标本身不出现在拉格朗日量L中。 其速度未必消失。
Do not discard the time derivative of a changing r squared factor.
不要丢弃r平方因子的时间导数。
holonomic constraint: A constraint expressible as a relation among coordinates and possibly time.
完整约束：可表示为坐标及可能时间之间关系的约束。
cyclic coordinate: A coordinate absent explicitly from the Lagrangian, with conserved canonical momentum under the Euler–Lagrange equation.
循环坐标：在拉格朗日量中不显式出现的坐标，其在欧拉-拉格朗日方程下具有守恒的共轭动量。
Retrieve the meanings without reading the definitions, then change a parameter in the original transfer and test the limiting case.
在不阅读定义的情况下回忆其含义，然后更改原始传递中的一个参数并检验极限情况。