Dynamics of point particle system 质点系动力学 Center of mass, center of mass frame Define center of mass Total momentum

m For continuum mass distribution Example: center of mass for a semicircle

center of mass frame 以质心为原点 Total momentum Total angular momentum 假设作用力沿着两个质点的连线 则上式为零

In center of mass frame Consider the relation of work and energy

If internal force are conservative If external force are conservative Total mechanical energy is conserved

Kinetic energy in center of mass frame

Tsiolkovsky rocket equation In vacuum Total mass M, dry mass rM Single stage two stage

Collision 碰撞 examples Fusion Neutron decay Large Hadron Collider:

Elastic collision 弹性碰撞， Q=0 Exothermic reaction 放热反映， Q>0 Endothermic reaction 吸热反应， Q<0 Fusion

1D elastic Collision Approaching and separation velocity or

静止靶 如果取粒子 1 静止，则粒子 2 反冲

1D inelastic Collision Complete inelastic collision 完全非弹性碰撞

2D elastic Collision Momentum and energy conservation provide 3 equations 2 final velocities are 4 unknowns. Depend on the detail of interaction. Assume impulse along unit vector n, then we have extra relations Elastic collision of two balls n

可以忽略相对论效应

Fluid dynamics 流体力学 Velocity field 速度场 Density distribution 密度分布 streamline

香蕉球 机翼的升力

Symmetry and conservation law Space-time, intrinsic symmetry Continuous, discrete symmetry Any global continuous symmetry has a corresponding conservation law Local symmetry, gauge symmetry Symmetry transformations depend on the space-time coordinate. Lorentz invariance General covariance Non-abelian gauge symmetry Special relativity General relativity Yang-Mills theory