- #1
mjordan2nd
- 177
- 1
In McCauley's book Classical Mchanics: Transformations, Flows, Integrable and Chaotic Dynamics we are analyzing a coordinate transformation in order to arrive at symmetry laws. A coordinate transformation is given by [itex]q_i(\alpha) = F_i(q_1,...,q_f, \alpha)[/itex]. Then, to the first order Mccauley states in equation 2.50 an infinitesimal shift in the coordinates can be given by
[tex]
\delta q_i = \left[ \frac{\partial q_i(\alpha)}{\partial \alpha} \right]_{\alpha=0} \delta \alpha
[/tex]
The variation in action is then given as
[tex] \delta A = \int_{t_1}^{t_2} \frac{dL_{\alpha}}{d \alpha} \delta \alpha dt = \int_{t_1}^{t_2} \left(\frac{\partial L_{\alpha}}{\partial q_i(\alpha)}{\partial q_i(\alpha)}{\partial \alpha} + \frac{\partial L_{\alpha}}{\partial \dot{q}_i(\alpha)} \frac{\partial \dot{q}_i(\alpha)}{\alpha} \right) \delta \alpha dt.[/tex]
I understand up to here. The book then states that we can deduce the following from the above:
[tex] \delta A = \left[ \left[ \frac{\partial L_{\alpha}}{\partial \dot{q}_i(\alpha)} \delta q_i \right]^{t_2}_{t_1} \right]_{\alpha=0} = \left[ p_i(\alpha) \left[\frac{\partial q_i(\alpha)}{\partial \alpha} \right]_{\alpha = 0} \delta \alpha \right]_{t_1}^{t_2} .[/tex]
I don't understand this line. For instance, shouldn't it be [itex] \delta \dot{q}_i[/itex] in the second expression? Or shouldn't the q be undotted? But then where does the second term from the above equation go? Again, any help would be appreciated.
Thanks.
[tex]
\delta q_i = \left[ \frac{\partial q_i(\alpha)}{\partial \alpha} \right]_{\alpha=0} \delta \alpha
[/tex]
The variation in action is then given as
[tex] \delta A = \int_{t_1}^{t_2} \frac{dL_{\alpha}}{d \alpha} \delta \alpha dt = \int_{t_1}^{t_2} \left(\frac{\partial L_{\alpha}}{\partial q_i(\alpha)}{\partial q_i(\alpha)}{\partial \alpha} + \frac{\partial L_{\alpha}}{\partial \dot{q}_i(\alpha)} \frac{\partial \dot{q}_i(\alpha)}{\alpha} \right) \delta \alpha dt.[/tex]
I understand up to here. The book then states that we can deduce the following from the above:
[tex] \delta A = \left[ \left[ \frac{\partial L_{\alpha}}{\partial \dot{q}_i(\alpha)} \delta q_i \right]^{t_2}_{t_1} \right]_{\alpha=0} = \left[ p_i(\alpha) \left[\frac{\partial q_i(\alpha)}{\partial \alpha} \right]_{\alpha = 0} \delta \alpha \right]_{t_1}^{t_2} .[/tex]
I don't understand this line. For instance, shouldn't it be [itex] \delta \dot{q}_i[/itex] in the second expression? Or shouldn't the q be undotted? But then where does the second term from the above equation go? Again, any help would be appreciated.
Thanks.