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Hamilton's Equations
Consider a dynamical system withgeneralized coordinates , for degrees of freedom which is described by the. Suppose that neither the kinetic energy, , nor the potential energy,systems, the potential energy is generally independent of the , depend explicitly on the time,. Now, in conventional dynamical , whereas the kinetic energy takes the form of a homogeneous quadratic function of the. In other words, (744) where theequation that depend on the , but not on the. It is easily demonstrated from the above (745) Recall, from Section 9.8, that generalized momentum conjugate to thecoordinate is defined th generalized (746) where is the Lagrangian of the system, and we have made use of the fact that is independent of the. Consider the function (747) If all of the conditions discussed above are satisfied then Equations (745) and (746) yield (748)
In other words, the functionConsider the variation of the function is equal to the. We have total energy of the system.
(749)
The first and third terms in the bracket cancel, because. Furthermore, since Lagrange's equation can be written (see Section 9.8), we obtain (750) Suppose, now, that we can express the total energy of the system, and the , with no explicit dependence on the. In other words, suppose that we can , solely as a function of the write Hamiltonian (^) of the system. The variation of the Hamiltonian function takes the form. When the energy is written in this fashion it is generally termed the (751) A comparison of the previous two equations yields (752) (753)
forHamilton's equations are often a useful alternative to Lagrange's equations, which take the. These first-order differential equations are known as Hamilton's equations. form ofConsider a one-dimensional harmonic oscillator. The kinetic and potential energies of the second-order differential equations. system are written and , where is the displacement, the mass, and. The generalized momentum conjugate to is (754)
Thus, the Hamiltonian of the system takes the form (763) In this case, Hamilton's equations yield (764) (765) which are just restatements of Equations (760) and (761), respectively, as well as (766) (767) The last equation implies that (768)
where is a constant. This can be combined with Equation (766) to give (769)
wheremotion for a particle moving in a central potential--see Chapter 5.. Of course, Equations (768) and (769) are the conventional equations of
Next: Richard Fitzpatrick 2011-03-31 Exercises Up: Hamiltonian Dynamics Previous: Constrained Lagrangian Dynamics
