62 lines
2.5 KiB
Markdown
62 lines
2.5 KiB
Markdown
---
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title: Motion in one dimension
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---
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The motion of a system having one degree of freedom is said to take place in one dimension. The most general form of the Lagrangian of such a system in fixed external conditions is
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L = 1a(q)i2-U(q),
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```load
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LL1/11.1
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```
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where $a(q)$ is some function of the generalised co-ordinate $q$. In particular,
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if $q$ is a Cartesian co-ordinate ($x$, say) then
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```load
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```
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The equations of motion corresponding to these Lagrangians can be integrated in a general form. It is not even necessary to write down the equation of motion; we can start from the first integral of this equation, which gives the law of conservation of energy. For the Lagrangian `LL1/11.2` (e.g.) we have $\mfrac{1}{2}m\dot{x}^2+U(x)=E$. This is a first-order differential equation, and can be inte- grated immediately. Since $\dd{x}/\dd{t} = \sqrt{2[E - U(x)]/m}$, it follows that
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```load
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LL1/11.3
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```
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The two arbitrary constants in the solution of the equations of motion are
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here represented by the total energy E and the constant of integration.
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Since the kinetic energy is essentially positive, the total energy always
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exceeds the potential energy, i.e. the motion can take place only in those
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regions of space where U(x) < E. For example, let the function U(x) be
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of the form shown in Fig. 6 (p. 26). If we draw in the figure a horizontal
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line corresponding to a given value of the total energy, we immediately find
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the possible regions of motion. In the example of Fig. 6, the motion can
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occur only in the range AB or in the range to the right of C.
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The points at which the potential energy equals the total energy,
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U(x) = E,
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(11.4)
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give the limits of the motion. They are turning points, since the velocity there
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is zero. If the region of the motion is bounded by two such points, then the
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motion takes place in a finite region of space, and is said to be finite. If the
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region of the motion is limited on only one side, or on neither, then the
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motion is infinite and the particle goes to infinity.
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A finite motion in one dimension is oscillatory, the particle moving re-
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peatedly back and forth between two points (in Fig. 6, in the potential well
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AB between the points X1 and x2). The period T of the oscillations, i.e. the
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time during which the particle passes from X1 to X2 and back, is twice the time
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from X1 to X2 (because of the reversibility property, §5) or, by (11.3),
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T(E) =
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(11.5)
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where X1 and X2 are roots of equation (11.4) for the given value of E. This for-
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mula gives the period of the motion as a function of the total energy of the
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particle.
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U
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A
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B
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C
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U=E
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x,
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X2
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X
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FIG. 6
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