How do I check if a transformation is a point transformation?

Question

In Lagrangian mechanics, I came across the notion of a point transformation which leaves the Lagrangian invariant. Normally it is denoted as follows.

$$Q = Q(q,t).$$

Now, unlike in the case of a canonical transformations (wherein there exist certain explicit conditions to check if a given transformation is canonical), I am unable to find a mechanism with which I can check whether a transformation, given as above, is a point transformation. Further, the instructor in my course said

'Unlike the Lagrangian which is invariant under any transformation, the Hamiltonian is not invariant under any arbitrary transformation. This is because the canonical coordinates and the conjugate momenta are independent in the Hamiltonian paradigm, but they are related in the Lagrangian paradigm.'

Does this mean that any transformation of the form $ Q = Q(q,t) $ is a point transformation i.e leaves the Lagrangian invariant? If not, how do I check if a given transformation is a point transformation?

Answer 1

Yes, a point transformation
$$q^i \longrightarrow Q^i~=~f^i(q,t)\tag{1}$$ is a canonical transformation (CT) with a type 2 generating function $$ F_2(q,P,t)~=~\sum_{i=1}^nf^i(q,t)P_i, \tag{2}$$

In the context of the Lagrangian formalism, it is usually argued that we have absolute freedom to perform point transformations.

Remember that Lagrange's equation are unchanged by point transformations.

Quick exercise for which would be very helpful -

Show that a point transformation is a canonical transformation.

Answer 2

1. A transformation $Q^j =f^j(q,\dot{q}, t)$ of generalized coordinates in Lagrangian mechanics is by definition called a point transformation if it doesn't depend on generalized velocities $\dot{q}^k$.

2. OP essentially asked:

   Q: Does it leave the Lagrangian invariant?

   A: It is invariant from the perspective of a passive coordinate transformation/reparametrization. It is generically not invariant from the perspective of an active transformation nor is it form invariant. (It should perhaps be stressed that Lagrangian invariance is irrelevant for the definition of a point transformation.)