三種不同的古典力學詮釋
Newtonian
Lagrangian
Hamiltonian
Newtonian (牛頓力學)
Newtonian 的變數是 F, x, x_dot, x_dot_dot; 2nd order differential equation
Newtonian 哲學是力造成速度的變化，正比於加速度
Pro: 1. 可以處理 non-conservative force (非保守場之力); 如摩擦力
2. 比較直覺
Con: 1. 力很難分析 (vector 而非 scaler; 有方向性)
2. 不同座標的 transformation 更複雜
Lagrangian
Lagrangian 的變數是 q and q_dot; 2nd order differential equation
Lagrangian 哲學是數小作用力原理
Pro: 1. L = T - U ; scaler only, no vector. Or use Newtonian to derive L.
2. General coordinates
Con: 1. 只處理 conservative force (保守場之力)
Hamiltonian
Hamiltonian 的變數是 p and q; 1nd order differential equation
Hamiltonian 哲學是 reversibility -> phase diagram -> fluid/flow Lioville equation
-> field theory
Pro: 1. H = Pq - L ; scaler only, no vector.
2. General coordinates
3. pq are symmetry
4. direct link to Quantum mechanics
Con: 1. 只處理 conservative force (保守場之力)
