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---------- (1a) |
the path of q(t); S is called the "functional" of q(t). In other words, a single value of S is evaluated from a path of q(t). It is sometimes written as S[q(t)] to emphasize its functional dependence of q(t). Figure 01a1 shows three different paths of q(t) (blue, green, red), and the corresponding Lagrangian L, which is also determined by the path of q(t). The action S[q(t)] would have three different numbers equal to the areas under L indicated by the blue, green or the red curve. | ||
Figure 01a1 _{} Functional of Paths |
Figure 01a2 S[q(t)] Functional of Paths _{} |
Figure 01a2 shows a set of more specific paths represented by the formula q = bt - (a/2)t^{2} with corresponding value of S[q(t)] (a ~ acceleration, b ~ velocity). |
---------- (1b) |
- L = constant ---------- (1g) |
which is the conservation of total energy (kinetic energy + potential energy) for a particle moving in a potential V(x). The momentum is not conserved in this case. However the dependence on x is eliminated in the free field case where V(x) = constant. It follows from Eq.(1h) that dx/dt = constant. This is the Newton's first law, which states that the object would not experience acceleration if there is no external force acting on it. Thus both the momentum m(dx/dt) and energy (m/2)(dx/dt)^{2} are conserved; or in term of symmetry, the system is now independent of time and space. Figure 01b shows schematically the similarity and difference between geometrical and theoretical physics symmetries. By symmetry, they both mean something is unchanged after some sort of rearrangement. However, in the case of geometrical symmetry it is the configuration that remains the same after the operation. Whereas in | |
Figure 01b Symmetry |
theoretical physics the invariance is about the form of the equations before and after the transformation. Although the laws of nature may seem to be simple and symmetrical, the real world isn't. It is messy |