- #1

gionole

- 281

- 24

1. Here , It says the following:

I wonder what is meant by independent ? When we vary the path ##q(t) -> q(t) + \epsilon k(t)## in the variational calculus, we also say that when that happens, ##\dot q(t)## changes and becomes ##\dot q(t) + \dot k(t)##. Could you provide an example of the counter argument that how it would look if they were not independent and what wouldn't work ?The variable ##\dot{q}## is **independent** of the variables ##t## and ##q## otherwise the derivative

##\frac{\partial L(t,q, \dot{q})}{\partial \dot{q}}## would be of dubious interpretation. The relation between ##q##, ##t##, and ##\dot{q}## is given by the second equation ##\frac{dq}{dt}= \dot{q}## and **it is valid only along the solution of the equations we are looking for.** The variables appearing in ##L(t,q,\dot{q})## are **independent of each other** before imposing the EL equations.

2. My question is now why 𝑓 can be dependent of ##\dot q## since only functions of the type ##\frac{d}{dt} f(q,t) leave the EL equations invariant ? Basically this is the same question as on the link, but I couldn't understand the answer.