Generalised energy and energy (lagrangian mechanics)

[supaman5]

I'm doing lagrangian mechanics and trying to understand my notes, are these three statements correct:

1. If kinetic energy is quadratic then energy equals generalised energy.
2. Saying kinetic energy of a system is quadratic is the same as saying none of the position vectors in a system depend explicitly on time.
3. A lagrangian can be independent of time (generalised energy is conserved) even if kinetic energy is not quadratic (energy DOESN'T equal generalised energy)

 

[supaman5]

Hopefully putting the formulas for generalised energy and energy might help:

L=T-V (kin-pot)

generalised energy: h= \sum\frac{\partial L}{\partial \dot{q}}\dot{q}-L where \dot{q} should be bold and is the time derivative vector for all generalised coordinates.

and energy: E=T+V

Took me about an hour to write out that formula...FAIL.