By Walter E Thirring
Mathematical Physics, Nat. Sciences, Physics, arithmetic
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Additional resources for A Course in Mathematical Physics, Vol. 1: Classical Dynamical Systems
21) the coordinates of the centre of mass do not appear. It follows that the relative motion occurs as if it were in accordance with the Lagrangian (where ), formed in exactly the same way as the Lagrangian for a single mass m equal to This mass is called the reduced mass. The motion of the centre of mass does not affect the relative motion of the masses. In particular, we can consider, simply, that the centre of mass lies at the coordinate origin R = 0. 19)], where . Then, if we describe the relative motion in spherical coordinates, the equations of motion will have the same form as for a single particle moving relative to a fixed centre of attraction.
Bodies in contact also give rise to forces of interaction which can be reduced to the kinematic properties of rigid constraints. If rigid constraints act in a system they force the particles to move on definite surfaces. Thus, in Sec. 1 we considered the motion of a single particle on a sphere, at the centre of which was another particle. This kind of interaction between particles does not cause a transition of the motion to the internal, microscopic, degrees of freedom of bodies. In other words, motion which is limited by rigid constraints is completely described by its own macroscopic generalized coordinates qα.
Hence, in a vacuum, in the absence of air resistance, all bodies fall with the same acceleration. Inertial frames of reference. 1) we have to do with the acceleration of a particle. There is no sense in talking about acceleration without stating to which frame of reference it is referred. For this reason there arises a difficulty in stating the cause of the acceleration. This cause may be either interaction between bodies or it may be due to some distinctive properties of the reference frame itself.
A Course in Mathematical Physics, Vol. 1: Classical Dynamical Systems by Walter E Thirring