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1 CHAPTER 14 HAMILTONIAN MECHANICS 14.1 Introduction The hamiltonian equations of motion are of deep theoretical interest. Having established that, I am bound to say that I have not been able to think of a problem in classical mechanics that I can solve more easily by hamiltonian methods than by newtonian or lagrangian methods. That is not to say that real problems cannot be solved by hamiltonian methods. What I have been looking for is a problem which I can solve easily by hamiltonian methods but which is more difficult to solve by other methods. So far, I have not found one. Having said that, doubt not that hamiltonian mechanics is of deep theoretical significance. Having expressed that mild degree of cynicism, let it be admitted that Hamilton theory – or more particularly its extension the Hamilton-Jacobi equations does have applications is celestial mechanics, and of course hamiltonian operators play a major part in quantum mechanics, although it is doubtful whether Sir William would have recognized his authorship in that connection. 14.2 A Thermodynamics Analogy Readers may have noticed from time to time – particularly in Chapter 9 that I have perceived some connection between parts of classical mechanics and thermodynamics. I perceive such an analogy in developing hamiltonian dynamics. Those who are familiar with thermodynamics may also recognize the analogy. Those who are not can skip this section without seriously prejudicing their understanding of subsequent sections. Please do not misunderstand: The hamiltonian in mechanics is not at all the same thing as enthalpy in thermodynamics, even though we use the same symbol, H . Yet there are similarities in the way we can introduce these concepts. In thermodynamics we can describe the state of the system by its internal energy, defined in such a way that when heat is supplied to a system and the system does external work, the increase in internal energy of the system is equal to the heat supplied to the system minus the work done by the system: . dV P dS T dU = 14.2.1 From this point of view we are saying that the internal energy is a function of the entropy and the volume: ) , ( V S U U = 14.2.2
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This note was uploaded on 10/25/2010 for the course MECHANIC Mechanic taught by Professor Monfered during the Fall '10 term at MIT.

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