Lagrangian Dynamics Homework: Find Missing Term

[davon806]

Homework Statement

Homework Equations

The last part of this question is an example of this result:

The Attempt at a Solution

Here is the solution

I think L' is missing a term: If we take the Earth as your frame of reference.(i.e. You are stationary, watching the movement of the railway carriage).Then there should be an extra term for the KE of pendulum,due to horizontal movement of the carriage. (see below, the y dot term corresponds to v of the carriage)

Why is it not involved in L' ?

 

[Orodruin]

Your constraint equation is $\dot y = v(t)$. This implies that the term you are referring to only depends on t and therefore is a total derivative (and hence irrelevant for the equations of motion).

 

[davon806]

Your constraint equation is $\dot y = v(t)$. This implies that the term you are referring to only depends on t and therefore is a total derivative (and hence irrelevant for the equations of motion).

Sorry, could you explain it in more detail? I couldn't get it

 

[Orodruin]

Which part do you have trouble with? There is only one degree of freedom for the pendulum, the angle $\theta$. The final Lagrangian cannot depend on $y$ since the motion of $y$ is given by integrating $v(t)$. The constraint gives you $\dot y = v(t)$ and so $y(t)$ is a primitive function of $v(t)$. Inserting the constraint into your Lagrangian means your $\dot y^2$ term turns into
$$
\frac{\mu v(t)^2}{2}.
$$
This is a function of $t$ only and does not affect the equations of motion.