0
votes
2answers
63 views

Taking time derivative of two dependant variables

I'm not entirely sure if this is correct. I have to take the time derivative of the following: $$\frac{d}{dt}mr^{2}\dot{\phi}$$ Now, both $r$ and $\dot{\phi}$ depends on the time $t$, so I have to ...
10
votes
2answers
360 views

Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $

I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are ...
8
votes
1answer
243 views

When motion begins, do objects go through an infinite number of position derivatives?

This might be a very vague and unclear question, but let me explain. When an object at rest moves, or moves from point $A$ to point $B$, we know the object must have had some velocity (1st derivative ...
1
vote
1answer
187 views

Clarification on a Goldstein formula steps (classical mechanics)

At page 20 of Classical Mechanics' Goldstein (Third edition), there are these two steps given between eqs. (1.51) and (1.52): $$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial ...
1
vote
2answers
242 views

What does $\textbf{f} = -\boldsymbol{\nabla} u$ mean in practice and how is it computed?

In classical computer simulations such as molecular dynamics (MD) simulations, one integrates Newton's equations of motion to determine particle trajectories. If we think of Newton's Second Law as ...