Hello professor, at 22:30, the three terms in the brackets are all zeros. Does this mean that dq/dt and dQ/dt are independent? Since Q = Q(q, p, t), it seems to me that q and Q are not independent, so that dq/dt and dQ/dt may not be independent. If so, how can we claim that the terms in the three brackets are all zeros? Thank you.
I think the idea is that in the Hamiltonian formalism, you think of the phase space s containing 2n independent variables to start with (q1 ... qn ; p1 ... pn) = (q, p). When you make a canonical transformation, you are essentially using a different coordinate system for phase space, (Q, P) instead of (q, p). However, in some circumstances the set (q, Q) may be enough to be a complete coordinate system on phase space, and from that perspective they are independent.
You are such an amazing teacher. Thank you so much!!
Wow, thank you!
Great Lectures with simplicity.
Glad you like them!
VERY CLEAR!!!
Thank you !
PLA requires only q to be fixed at t1 and t2, but qdot is not fixed. How can we make sure that p,Q,P are fixed at t1and t2?
Please give an example of the relationship between the Lagrangian and a generating function
What did you do around 11:45 ? How do you come to that result from that integral
Distribute the variation $\delta$ operator into the integral, and then into the time derivative, where you end up with:
$$\delta \int_{t_1} ^{t_2} \frac{d}{dt} F(q, Q, t) dt = \int_{t_1} ^{t_2} \delta \frac{d}{dt} F(q, Q, t) dt = \int_{t_1} ^{t_2} \frac{d}{dt} \delta F(q, Q, t) dt$$
Expand using the chain rule, where:
$$\delta F(q, Q, t) = \frac{\partial F}{\partial q} \delta q + \frac{\partial F}{\partial Q} \delta Q$$
$$\implies \int_{t_1} ^{t_2} \frac{d}{dt} \delta F(q, Q, t) dt = \int_{t_1} ^{t_2} \frac{d}{dt} \frac{\partial F}{\partial q} \delta q + \frac{\partial F}{\partial Q} \delta Q dt$$
$$\implies \frac{\partial F}{\partial q} (q(t_2), Q(t_2), t_2) \delta q(t_2) + \frac{\partial F}{\partial Q} (q(t_1), Q(t_1), t_1) \delta Q(t_1)$$
Hello professor, at 22:30, the three terms in the brackets are all zeros. Does this mean that dq/dt and dQ/dt are independent? Since Q = Q(q, p, t), it seems to me that q and Q are not independent, so that dq/dt and dQ/dt may not be independent. If so, how can we claim that the terms in the three brackets are all zeros? Thank you.
I think the idea is that in the Hamiltonian formalism, you think of the phase space s containing 2n independent variables to start with (q1 ... qn ; p1 ... pn) = (q, p). When you make a canonical transformation, you are essentially using a different coordinate system for phase space, (Q, P) instead of (q, p). However, in some circumstances the set (q, Q) may be enough to be a complete coordinate system on phase space, and from that perspective they are independent.