Energy change under point transformation

Question

How do the energy and generalized momenta change under the following coordinate

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

The new momenta: $$P = \partial L / \partial \dot Q = \partial L / \partial \dot q\times \partial \dot q / \partial \dot Q = p \partial \dot q / \partial \dot Q = p \partial q / \partial Q. $$

The new velocity:

$$\dot Q = \partial Q / \partial q \times \dot q + \partial Q / \partial t.$$

The new energy:

$$E' = P\dot Q - L = p \partial q / \partial Q (\partial Q / \partial q \times \dot q + \partial Q / \partial t) - L = p\dot q + p \partial q / \partial t - L = E + p \partial q / \partial t.$$

But the answer is $$E' = E - p \partial q / \partial t.$$

What did i got wrong?

Answer 1

Hint: The triple product rule: $$ \left(\frac{\partial q}{\partial Q}\right)_t\left(\frac{\partial Q}{\partial t}\right)_q ~=~-\left(\frac{\partial q}{\partial t}\right)_Q.$$