Central force energy conservation

If one adopts McLennan s [78b] interpretation, then Eq. (21) is a realization of a standard theorem of Newtonian mechanics conservation of total energy = conservation of kinetic plus potential energy (see, e.g., Chap. 4 of Kleppner and Kolenkow, [80]). The reason is simple Coulomb electric force is central, then work is path independent, and total energy is function of position only. The time derivative of total energy is of course zero, as in Eq. (21). In this interpretation Qp and Qi are manifestations of kinetic energy. 

For conservative central forces and a defined interaction potential, V(r), we can write a statement for the total mechanical energy for a particle of mass M, a distance r away from a central force F as... 

In classical mechanics a particle subject to a central force has its angular momentum conserved (Section 5.3). In quantum mechanics we might ask whether we can have states with definite values for both the energy and the angular momentum. To have the set of eigenfunctions of H also be eigenfunctions of V-, the commutator [H, D] must vanish. We have... 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

The most celebrated textual embodiment of the science of energy was Thomson and Tait s Treatise on Natural Philosophy (1867). Originally intending to treat all branches of natural philosophy, Thomson and Tait in fact produced only the first volume of the Treatise. Taking statics to be derivative from dynamics, they reinterpreted Newton s third law (action-reaction) as conservation of energy, with action viewed as rate of working. Fundamental to the new energy physics was the move to make extremum (maximum or minimum) conditions, rather than point forces, the theoretical foundation of dynamics. The tendency of an entire system to move from one place to another in the most economical way would determine the forces and motions of the various parts of the system. Variational principles (especially least action) thus played a central role in the new dynamics. 

The system evolves with the innermost orbit receding from the central body (because of the non-conservative forces acting on mi) up to the moment where the system is captured into a resonance, a2 is almost constant. When the 2/f-resonance is reached, the system is trapped by the resonance. As known since Laplace, after the capture, mi continuously transfers one fraction of the energy that it is getting from the non-conservative source to m2, so that 02 also increases. One may note from Figure 9 that, after the capture into the resonance, ai increases at a smaller pace than before the capture. The increase of the semi-major axes is such that the ratio ai/a2 remains constant. 

From classical mechanics it follows that for an isolated system (and assum ing the forces to be central and obeying the action-reaction principle), its energy, momentum and angular momentum are conserved. 

Throughout this textbook we will be studying the Coulomb force interactions of various particles atomic nuclei and electrons, atoms and other atoms, molecules and other molecules. From the potential energy function and the total energy, we can in principle determine all the possible results of these interactions. Because the total energy is conserved, the potential energy becomes the key to many central problems throughout physical chemistry. Keep an eye on it. 

Provided no external forces are acting on the body, its total energy is constant. This remark is elevated to a central statement of classical physics known as the law of the conservation of energy. Potential and kinetic energy may be freely... 

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