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Physics
Classical Physics
Is energy always conserved in a co-rotating frame?
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[QUOTE="ergospherical, post: 6536301, member: 691758"] As I think I wrote in one of your previous threads, there is a simple form of energy conservation that applies to uniformly rotating, non-translationally-accelerating frames, viz: ##\dfrac{d}{dt} \left( T - \frac{1}{2}I \Omega^2 \right) - \displaystyle{\sum_a} \mathbf{F}_a \cdot \mathbf{v}_a = 0##. If all of the ##\mathbf{F}_a## are conservative then ##\displaystyle{\sum_a} \mathbf{F}_a \cdot \mathbf{v}_a## is a total time derivative and you have a conserved energy. For other systems, whether or not you can find energy integrals depends on whether there is time dependence in the lagrangian i.e. write ##H = \dot{q}^i \dfrac{\partial L}{\partial \dot{q}^i} - L## then if ##\partial L/\partial t=0## you have$$\dfrac{dH}{dt} = \dot{q}^i \dfrac{d}{dt} \dfrac{\partial L}{\partial \dot{q}^i} + \dfrac{\partial L}{\partial \dot{q}^i} \ddot{q}^i - \dfrac{\partial L}{\partial q^i} \dot{q}^i - \dfrac{\partial L}{\partial \dot{q}^i} \ddot{q}^i$$which equals zero. [/QUOTE]
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Classical Physics
Is energy always conserved in a co-rotating frame?
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