This question actually occurred in a Physics problem, however I thought it would be more appropriate to ask it here as it's only about the mathematical part.

The question was about a pendulum with an accelerated support point using Lagrangian mechanics. The Lagrangian involved the following term:


where $\dot{y}=\frac{1}{2}at^2$. When evaluating the RHS of the Euler-Lagrange equation $\frac{d}{dt}(\frac{\partial{L}}{\partial\dot\theta})=\frac{\partial{L}}{\partial\theta}$ I had the following problem:

How do I evaluate $\frac{\partial}{\partial\theta}(\frac{d\theta}{dt})$?

My instinct would tell me as $\theta$ is a function of t only that this would be equivalent to evaluating $\frac{d}{d\theta}(\frac{d\theta}{dt})$. Using the chani rule:


Therefore given the whole differentiated term using the product rule as:


However my textbook leaves out the second term and only takes the first term, i.e. it takes


Summarised my question is: How do I evaluate these two expressions:

$\frac{\partial}{\partial\theta}(\frac{d\theta}{dt})$ and $\frac{d}{d\theta}(\frac{d\theta}{dt})$

when $\theta=\theta(t)$


In the Euler–Lagrange equations, when computing $\partial L/\partial \theta$ and $\partial L/\partial \dot\theta$, the quantities $\theta$ and $\dot \theta$ are treated as independent variables, so $\frac{\partial}{\partial \theta} \dot \theta = 0$. At this stage, you shouldn't think of $\theta$ as depending on $t$. (That doesn't enter until you compute $\frac{d}{dt} (\partial L/\partial \dot\theta)$.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.