sum_j (v_j-V)^2 = sum_j ( (v_j+V') - (V+V') )^2 is an identity!

送交者: gyro 于 2009-10-29, 00:21:19:

回答: 说了半天，不就是要说明：“相对于质心的动能”与参照系无关吗？由 Palmetto 于 2009-10-29, 00:13:56:

v_j is the jth patticle velocity in a frame (frame A) in which the center-of-mass velocity is V;

If frame A moves at velocity V' relative to another frame (frame B), then the jth particle velocity becomes v_j+V'in frame B, the center-of-mass velocity in frame B becomes V+V'.

The identity says that the total kinetic energy relative to the center-of-mass frame is independent of the reference frame.

所有跟贴:

前提是存在一个确定的质心速度V - wasguru (32 bytes) 2009-10-29, 00:32:37 (378185)
- Nothing is assumed about dV/dt for the identity to be valid. (无内容) - gyro (0 bytes) 2009-10-29, 00:42:47 (378191)
这个等式的前题是v_j^2=V^2+(v_j-V)^2，我就是在证明这个。 (无内容) - Palmetto (0 bytes) 2009-10-29, 00:29:18 (378183)
- Palmetto, you must be twisting something weird in your mind - gyro (51 bytes) 2009-10-29, 00:37:02 (378188)
  - 速度为v_j的质点j的能量是v_j^2,质心速度是V,所以j的能量为v_j^2=V^2+(v_j-V)^2。 (无内容) - Palmetto (0 bytes) 2009-10-29, 00:50:19 (378193)
    - But v_j^2=V^2+(v_j-V)^2 is absurd and not needed - gyro (245 bytes) 2009-10-29, 00:57:54 (378195)
      - 另外，如果不对时间平均的话，v_j^2是个变数，非守恒。 - Palmetto (75 bytes) 2009-10-29, 02:04:48 (378216)
        
        Who said m_j v_j^2 is conserved? - gyro (363 bytes) 2009-10-29, 02:12:50 (378220)
        
        咱们不是在讨论用能量守恒证明动量守恒吗？不守恒如何证明？ - Palmetto (22 bytes) 2009-10-29, 02:46:47 (378228)
      - 你这样，不是提前用了动量守恒吗？ (无内容) - Palmetto (0 bytes) 2009-10-29, 01:56:17 (378212)
        
        I never take the Berkeley "proof" seriously - gyro (151 bytes) 2009-10-29, 02:05:00 (378217)
        
        仅仅定义是不够的，因为v_j是变数，它们的合可能也是。 (无内容) - Palmetto (0 bytes) 2009-10-29, 02:43:54 (378225)
      - v_j^2=V^2+(v_j-V)^2的成立，必须对时间平均。 (无内容) - Palmetto (0 bytes) 2009-10-29, 01:16:10 (378199)
      - 别弄那么多点了，只考虑用弹簧连接的两个球，两球振动v=a*coswt，质心速度为V。 - Palmetto (36 bytes) 2009-10-29, 01:08:03 (378196)

加跟贴