Editing Generalized coordinates (section)

===Generalized momentum===

The ''generalized momentum'' "[[canonical coordinates|canonically conjugate]] to" the coordinate {{mvar|q{{sub|i}}}} is defined by

:<math>p_i =\frac{\partial L}{\partial\dot q_i}.</math>

If the Lagrangian {{mvar|L}} does ''not'' depend on some coordinate {{mvar|q{{sub|i}}}}, then it follows from the Euler–Lagrange equations that the corresponding generalized momentum will be a [[conserved quantity]], because the time derivative is zero implying the momentum is a constant of the motion;

:<math> \frac{\partial L}{\partial q_i} =\frac{d}{dt}\frac{\partial L}{\partial\dot q_i} =\dot{p}_i =0\,.</math>