Lagrangians in generalized coordinates
The function
takes a coordinate path; the function
takes a configuration path.
![\displaystyle{\begin{aligned} \mathcal{S} [\gamma] (t_1, t_2) &= \int_{t_1}^{t_2} \mathcal{L} \circ \mathcal{T} [\gamma] \\ S_\chi [q] (t_1, t_2) &= \int_{t_1}^{t_2} L_\chi \circ \Gamma [q] \\ \end{aligned}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cbegin%7Baligned%7D++%5Cmathcal%7BS%7D+%5B%5Cgamma%5D+%28t_1%2C+t_2%29+%26%3D+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+%5Cmathcal%7BL%7D+%5Ccirc+%5Cmathcal%7BT%7D+%5B%5Cgamma%5D++%5C%5C+++S_%5Cchi+%5Bq%5D+%28t_1%2C+t_2%29+%26%3D+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+L_%5Cchi+%5Ccirc+%5CGamma+%5Bq%5D++%5C%5C+++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{\begin{aligned} \mathcal{S} [\gamma] (t_1, t_2) &= \int_{t_1}^{t_2} \mathcal{L} \circ \mathcal{T} [\gamma] \\ S_\chi [q] (t_1, t_2) &= \int_{t_1}^{t_2} L_\chi \circ \Gamma [q] \\ \end{aligned}}")
![\displaystyle{\begin{aligned} \mathcal{S} [\gamma] (t_1, t_2) &= S_\chi [\chi \circ \gamma] (t_1, t_2) \\ \end{aligned}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cbegin%7Baligned%7D++%5Cmathcal%7BS%7D+%5B%5Cgamma%5D+%28t_1%2C+t_2%29++%26%3D+S_%5Cchi+%5B%5Cchi+%5Ccirc+%5Cgamma%5D+%28t_1%2C+t_2%29+%5C%5C++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{\begin{aligned} \mathcal{S} [\gamma] (t_1, t_2) &= S_\chi [\chi \circ \gamma] (t_1, t_2) \\ \end{aligned}}")
Computing Actions
makes a procedure that represents a function of one argument that has no known properties other than the given symbolic name.
The method of computing the action from the coordinate representation of a Lagrangian and a coordinate path does not depend on the coordinate system.
Exercise 1.4. Lagrangian actions
For a free particle an appropriate Lagrangian is
Suppose that
is the constant-velocity straight-line path of a free particle, such that
and
. Show that the action on the solution path is
— Structure and Interpretation of Classical Mechanics
.
![\displaystyle{\begin{aligned} S_\chi [\gamma] (t_1, t_2) &= \int_{t_1}^{t_2} L_\chi (t, q(t), Dq(t)) dt \\ &= \int_{t_2}^{t_1} \frac{1}{2} m v^2 dt \\ &= \frac{1}{2} m v^2 \int_{t_2}^{t_1} dt \\ &= \frac{1}{2} m v^2 (t_2 - t_1) \\ &= \frac{1}{2} m (\frac{x_2 - x_1}{t_2 - t_1})^2 (t_2 - t_1) \\ &= \frac{1}{2} m \frac{(x_2 - x_1)^2}{t_2 - t_1} \\ \end{aligned}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B%5Cbegin%7Baligned%7D++S_%5Cchi+%5B%5Cgamma%5D+%28t_1%2C+t_2%29+%26%3D+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+L_%5Cchi+%28t%2C+q%28t%29%2C+Dq%28t%29%29+dt+%5C%5C++%26%3D+%5Cint_%7Bt_2%7D%5E%7Bt_1%7D+%5Cfrac%7B1%7D%7B2%7D+m+v%5E2+dt+%5C%5C++%26%3D+%5Cfrac%7B1%7D%7B2%7D+m+v%5E2+%5Cint_%7Bt_2%7D%5E%7Bt_1%7D+dt+%5C%5C++%26%3D+%5Cfrac%7B1%7D%7B2%7D+m+v%5E2+%28t_2+-+t_1%29++%5C%5C++%26%3D+%5Cfrac%7B1%7D%7B2%7D+m+%28%5Cfrac%7Bx_2+-+x_1%7D%7Bt_2+-+t_1%7D%29%5E2+%28t_2+-+t_1%29++%5C%5C++%26%3D+%5Cfrac%7B1%7D%7B2%7D+m+%5Cfrac%7B%28x_2+-+x_1%29%5E2%7D%7Bt_2+-+t_1%7D+++%5C%5C+++%5Cend%7Baligned%7D%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{\begin{aligned} S_\chi [\gamma] (t_1, t_2) &= \int_{t_1}^{t_2} L_\chi (t, q(t), Dq(t)) dt \\ &= \int_{t_2}^{t_1} \frac{1}{2} m v^2 dt \\ &= \frac{1}{2} m v^2 \int_{t_2}^{t_1} dt \\ &= \frac{1}{2} m v^2 (t_2 - t_1) \\ &= \frac{1}{2} m (\frac{x_2 - x_1}{t_2 - t_1})^2 (t_2 - t_1) \\ &= \frac{1}{2} m \frac{(x_2 - x_1)^2}{t_2 - t_1} \\ \end{aligned}}")
— Me@2006-2008
— Me@2019-03-10 11:08:29 PM
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2019.03.10 Sunday (c) All rights reserved by ACHK
