we have seen that the Eular-Lagrange equations are form invariant under an arbitrary change of coordinates in configuation space.[$\overline q:q=q(\overline q)$]
If the equations of motion are simplified by using a new set of generalized variables $(Q,P)$, compared to using the original set of variables $(q,p)$, then an advantage has been gained.
In Hamiltonian mechanics, a cononical transformation is a change of canonical coordinates $(q,p,r)\to (Q,P,t)$ that preseves the form of Hamilton’s equation.
$$ \mathbf{\dot{q}}=\frac{\partial H(\mathbf{q},\mathbf{p},t)}{\partial\mathbf{p}}\\-\mathbf{\dot{p}}=\frac{\partial H(\mathbf{q},\mathbf{p},t)}{\partial\mathbf{q}} $$
⇒
$$ \dot{\mathbf{Q}}=\frac{\partial\mathcal{H}(\mathbf{Q},\mathbf{P},t)}{\partial\mathbf{P}}\\-\dot{\mathbf{P}}=\frac{\partial\mathcal{H}(\mathbf{Q},\mathbf{P},t)}{\partial\mathbf{Q}} $$
There exists a function such that $P,Q$ are still governed by Hamilton’s equations, even the new functions may be very different from the old Hamiltonian.
$$ \{\overline q_j,\overline q_k\}=0=\{\overline p_j,\overline p_k\}\\\{\overline q_j,\overline p_k\}=\delta_{jk} $$
区别于拉格朗日量的规范不变性。
拉格朗日量的坐标变换coordinate transformations(point transformation) are a type of canonical transformation.
$$ \delta S=\delta\int_{t_1}^{t_2}L(\mathbf{q},\mathbf{\dot{q}},t)dt=\delta\int_{t_1}^{t_2}[\mathbf{p}\cdot\mathbf{\dot{q}}-H(\mathbf{q},\mathbf{p},t)]dt=0 $$
Applying Hamilton’s principle of least action to new Lagrangian $\mathcal{L}(\mathbf{Q},\dot{\mathbf{Q}},t)$ gives:
$$ \delta S=\delta\int_{t_1}^{t_2}\mathcal{L}(\mathbf{Q},\mathbf{\dot{Q}},t)dt=\delta\int_{t_1}^{t_2}\left[\mathbf{P}\cdot\mathbf{\dot{Q}}-\mathcal{H}(\mathbf{Q},\mathbf{P},t)\right]dt=0 $$
Using gauge-invariant Lagrangians $\frac{dF}{dt}+\mathcal L={L}$. Generating function F.