Energy change under point transformation

[LCSphysicist]

Homework Statement

I will put it below.

Relevant Equations

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How do the energy and generalized momenta change under the following coordinate
transformation $$q= f(Q,t)$$The generalized momenta: $$P = \partial L / \partial \dot Q = \partial L / \partial \dot q\times \partial \dot q / \partial \dot Q = p \partial \dot q / \partial \dot Q = p \partial q / \partial Q $$$$\dot Q = \partial Q / \partial q \times \dot q + \partial Q / \partial t$$$$E' = P\dot Q - L = p \partial q / \partial Q (\partial Q / \partial q \times \dot q + \partial Q / \partial t) - L = p\dot q + p \partial q / \partial t - L = E + p \partial q / \partial t$$But the answer is $$E' = E - p \partial q / \partial t$$What did i got wrong?

 

[TSny]

When writing a partial derivative, it's important to keep in mind which variables are held constant. I often find it necessary to explicitly indicate the variable or variables that are held constant.

I think the mistake occurs where you essentially wrote $$\frac{\partial q}{\partial Q} \frac{\partial Q}{\partial t} = \frac{\partial q}{\partial t} $$

Convince yourself that the left side should be $$\frac{\partial q}{\partial Q}\bigg |_t \frac{\partial Q}{\partial t}\bigg|_q $$ You can simplify this using the triple product rule (also called the cyclic identity for partial derivatives).