- #1
Happiness
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It seems like I could get the Euler-Lagrange equation for any function that allows symmetry of second derivatives even when the action is not stationary.
Suppose ##L=L(q_1, q_2, ... , q_n, \dot{q_1}, \dot{q_2}, ... , \dot{q_n}, t)##, where all the ##q_i##'s and ##\dot{q_i}##'s are functions of ##t##. ##(\dot{x}## represents ##\frac{dx}{dt}##.##)##
##\frac{d}{dt}\frac{\partial L}{\partial\dot{q}}=\Sigma_i\Big(\frac{dq_i}{dt}\frac{\partial}{\partial q_i}\frac{\partial L}{\partial\dot{q}}+\frac{d\dot{q_i}}{dt}\frac{\partial}{\partial\dot{q_i}}\frac{\partial L}{\partial\dot{q}}\Big)+\frac{\partial}{\partial t}\frac{\partial L}{\partial\dot{q}}##
##=\Sigma_i\Big(\frac{dq_i}{dt}\frac{\partial}{\partial\dot{q}}\frac{\partial L}{\partial q_i}+\frac{d\dot{q_i}}{dt}\frac{\partial}{\partial\dot{q}}\frac{\partial L}{\partial\dot{q_i}}\Big)+\frac{\partial}{\partial\dot{q}}\frac{\partial L}{\partial t}##
##=\frac{\partial}{\partial\dot{q}}\frac{dL}{dt}##
##=\frac{\partial\dot{L}}{\partial\dot{q}}##
##=\frac{\partial L}{\partial q}##
Therefore, ##\frac{d}{dt}\frac{\partial L}{\partial\dot{q}}-\frac{\partial L}{\partial q}=0##.
What's wrong?
EDIT: I think I found the mistake. ##\frac{\partial\dot{L}}{\partial\dot{q}}=\frac{\partial L}{\partial q}## when ##L=L(q, t)## but not so when ##L=L(q, \dot{q}, t)##.
Suppose ##L=L(q_1, q_2, ... , q_n, \dot{q_1}, \dot{q_2}, ... , \dot{q_n}, t)##, where all the ##q_i##'s and ##\dot{q_i}##'s are functions of ##t##. ##(\dot{x}## represents ##\frac{dx}{dt}##.##)##
##\frac{d}{dt}\frac{\partial L}{\partial\dot{q}}=\Sigma_i\Big(\frac{dq_i}{dt}\frac{\partial}{\partial q_i}\frac{\partial L}{\partial\dot{q}}+\frac{d\dot{q_i}}{dt}\frac{\partial}{\partial\dot{q_i}}\frac{\partial L}{\partial\dot{q}}\Big)+\frac{\partial}{\partial t}\frac{\partial L}{\partial\dot{q}}##
##=\Sigma_i\Big(\frac{dq_i}{dt}\frac{\partial}{\partial\dot{q}}\frac{\partial L}{\partial q_i}+\frac{d\dot{q_i}}{dt}\frac{\partial}{\partial\dot{q}}\frac{\partial L}{\partial\dot{q_i}}\Big)+\frac{\partial}{\partial\dot{q}}\frac{\partial L}{\partial t}##
##=\frac{\partial}{\partial\dot{q}}\frac{dL}{dt}##
##=\frac{\partial\dot{L}}{\partial\dot{q}}##
##=\frac{\partial L}{\partial q}##
Therefore, ##\frac{d}{dt}\frac{\partial L}{\partial\dot{q}}-\frac{\partial L}{\partial q}=0##.
What's wrong?
EDIT: I think I found the mistake. ##\frac{\partial\dot{L}}{\partial\dot{q}}=\frac{\partial L}{\partial q}## when ##L=L(q, t)## but not so when ##L=L(q, \dot{q}, t)##.
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