39 lines
1.5 KiB
Markdown
39 lines
1.5 KiB
Markdown
---
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title: 41-the-routhian
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---
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The Routhian
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133
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of the Lagrangian, since the time would there be only a parameter which
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would not be involved in the transformation. Analogously to formula (40.6),
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the partial time derivatives of L and H are related by
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(40.8)
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PROBLEMS
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PROBLEM 1. Find the Hamiltonian for a single particle in Cartesian, cylindrical and
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spherical co-ordinates.
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SOLUTION. In Cartesian co-ordinates x, y, 2,
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in cylindrical co-ordinates r, , z,
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in spherical co-ordinates r, 0, ,
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PROBLEM 2. Find the Hamiltonian for a particle in a uniformly rotating frame of reference.
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SOLUTION. Expressing the velocity V in the energy (39.11) in terms of the momentum p
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by (39.10), we have H = p2/2m-S rxp+U.
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PROBLEM 3. Find the Hamiltonian for a system comprising one particle of mass M and n
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particles each of mass m, excluding the motion of the centre of mass (see §13, Problem).
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SOLUTION. The energy E is obtained from the Lagrangian found in §13, Problem, by
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changing the sign of U. The generalised momenta are
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Pa = OL/OV
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Hence
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-
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= (mM/14)
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=
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=
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Substitution in E gives
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41. The Routhian
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In some cases it is convenient, in changing to new variables, to replace
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only some, and not all, of the generalised velocities by momenta. The trans-
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formation is entirely similar to that given in 40.
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To simplify the formulae, let us at first suppose that there are only two
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co-ordinates q and E, say, and transform from the variables q, $, q, $ to
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q, $, p, & where P is the generalised momentum corresponding to the co-
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ordinate q.
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134
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The Canonical Equations
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