Recent content by RohanJ

I forgot to ask one thing: For any path other than a geodesic, does the tangent vectors not form a parallel field,i.e., if I take the tangent vector at a point and parallel transport it to the next point, does it coincide with the tangent vector at the next point only for geodesics?

I am currently reading Foster and Nightingale and when it comes to the concept of parallel transport, the authors don't go very deep in explaining it except just stating that if a vector is subject to parallel transport along a parameterized curve, there is no change in its length or direction...

In Introduction to special relativity by Resnick,there is a thought experiment to compare lengths perpendicular to relative motion as given in the below image. What if we try to perform such an experiment to compare lengths parallel to relative motion? Suppose there are two horizontal rods...

Yes and that was my only concern that why should ##\ddot q## be independent..

Okay,got it now. Now I will do the second part to confirm the concept. Thank you very much.

We don't know anything about ##\ddot q##. How do we know it is an independent variable? Can you provide any text references for further reading into this topic?

I don't see how it is straightforward. Can you elaborate?

##\frac{df}{dt}## will be equal to ##\frac{\partial f}{\partial t} + \frac{\partial f}{\partial q}\frac{dq}{dt} + \frac{\partial f}{\partial \dot q}\frac{d\dot q}{dt}## ,I think.

##\frac{d}{dt}## means differentiating with respect to time. I have used this definition to get the results I got.

I don't understand what you asked about the definition of ##\frac{d}{dt}## but here is the question:

Since ##\dot q , q## and ##t ## are independent variables ## \frac{\partial }{\partial q}\frac {dq}{dt}## is zero but what about the term ##\frac{\partial}{\partial q}\frac{d \dot q}{dt}##?

Problem Statement: Use the definition of the total time derivative to a) show that ##(∂ /∂q)(d/dt)f(q,q˙,t) = (d /dt)(∂/∂q)f(q,q˙,t)## i.e. these derivatives commute for any function ##f = f(q, q˙,t)##. Relevant Equations: My approach is given below. Please tell if it is correct and if not ...

Sorry I made a mistake.I meant why the term containing ##∂L/∂v^2## or the "second term" i.e. ## 2v.e(∂L/∂v^2)## must be linear in velocity?

I understand that addition of a total time derivative does not affect the Euler Lagrange equation. My question is in the discussion in Landau mechanics ,why does the second term of the taylor expansion being the total time derivative leads to the conclusion that ##∂L/∂v^2## is a linear function...

I don't understand why the second term must be linear in velocity and the part after that.