Suppose two observers A and B are each accelerating at the same rate wrt an inertial reference frame, but are moving relative to one another at some uniform velocity. Will the transformation between the coordinates of an event as measured by A and B be the Lorentz transformation?
Does anyone have an English translation of Einstein's 1906 paper "The Principle of Conservation of Motion of the Center of Gravity and the Inertia of Energy"? A copy would be greatly appreciated.
We all know the proof, from Newtonian mechanics, that the motion of the center of mass of a system of particles can be found by treating the center of mass as a particle with all the external forces acting on it. I want to prove the same think, but within the framework of Lagrangian mechanics...
Yes. Especially because I haven't yet learned the Hamiltonian formalism. I would think that there's a more or less elementary way to show \dot{x}^2 + x^2 + \frac{\epsilon}{2}x^4 =\mathrm{constant} closes on itself, but try as I might I can't come up with anything...
You're absolutely right. I guess the definition for an equilibrium configuration is that \ddot{q}_j=0. I'll have to go back to the drawing board...
Thanks for your help!
In general, how do you prove that a given trajectory in phase space is closed?
For example, suppose the energy E of a one-dimensional system is given by E=\frac{1}{2}\dot{x}^2 +\frac{1}{2}x^2 + \frac{\epsilon}{4}x^4, where ε is a positive constant. Now, I can easily show that all phase...
Fi and ri are the total force of the ith particle and the position of the ith particle, respectively. I introduced force because equilibrium, by definition, occurs when all the forces on the particles vanish. I need to bring that in somewhere.
And yes, you've defined scleronomic correctly.
No one has answered my question. I can only assume that I was unclear in formulating it. So please, if there's something in my post that is confusing, let me know so I can clarify what I'm trying to ask.
Suppose we have a system with scleronomic constraints. Is the condition that ∂V/∂qj=0 for generalized coordinates qj a necessary condition for equilibrium? A sufficient condition?
I managed to "prove" that the above condition is necessary and sufficient for any type of holonomic constaint...
I hate bumping my own question, but this question is really bugging me. If you need me to clarify some points, or don't understand what I've written at all, please let me know.
To add a little more detail:
It's necessarily the case that there's a flaw in my proof, because I can find an example of a rheonomic system where the equilibrium points don't satisfy \frac{\partial V}{\partial q_j}=0. Take a point mass on a vertically oriented hoop of radius R rotating w/...