34 lines
2.9 KiB
Markdown
34 lines
2.9 KiB
Markdown
---
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title: Generalized co-ordinates
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---
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One of the fundamental concepts of mechanics is that of a particle[^1]. By this
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we mean a body whose dimensions may be neglected in describing its motion.
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The possibility of so doing depends, of course, on the conditions of the prob-
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lem concerned. For example, the planets may be regarded as particles in
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considering their motion about the Sun, but not in considering their rotation
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about their axes.
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The position of a particle in space is defined by its radius vector $r$, whose
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components are its Cartesian co-ordinates $x$, $y$, $z$. The derivative $\v{v} = \dd{\v{r}}/\dd{t}$. of $r$ with respect to the time $t$ is called the velocity of the particle, and the second derivative $\dd[2]{\v{r}}/\dd[2]{t}$ is its acceleration. In what follows we shall, as is customary, denote differentiation with respect to time by placing a dot above a letter: $\v{v} = \dot{\v{r}}$.
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To define the position of a system of N particles in space, it is necessary to
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specify $N$ radius vectors, i.e. $3N$ co-ordinates. The number of independent
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quantities which must be specified in order to define uniquely the position of
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any system is called the number of degrees of freedom; here, this number is $3N$. These quantities need not be the Cartesian co-ordinates of the particles,
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and the conditions of the problem may render some other choice of co-
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ordinates more convenient. Any $S$ quantities $q_1$, $q_2$, ..., $q_s$ which completely define the position of a system with $S$ degrees of freedom are called generalised co-ordinates of the system, and the derivatives $q_i$ are called its generalised velocities.
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When the values of the generalised co-ordinates are specified, however, the "mechanical state" of the system at the instant considered is not yet determined in such a way that the position of the system at subsequent instants can be predicted. For given values of the co-ordinates, the system can have any velocities, and these affect the position of the system after an infinitesimal time interval $\dd{t}$.
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If all the co-ordinates and velocities are simultaneously specified, it is
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known from experience that the state of the system is completely determined
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and that its subsequent motion can, in principle, be calculated. Mathematic-
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ally, this means that, if all the co-ordinates $q$ and velocities $\dot{q}$ are given at some instant, the accelerations $\ddot{q}$ at that instant are uniquely defined[^2].
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The relations between the accelerations, velocities and co-ordinates are
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called the equations of motion. They are second-order differential equations
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for the functions $q(t)$, and their integration makes possible, in principle, the determination of these functions and so of the path of the system.
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[^1]: Sometimes called in Russian a material point
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[^2]: For brevity, we shall often conventionally denote by q the set of all the co-ordinates $q_1,q_2,...,q_s$ and similarly by $\dot{q}$ the set of all velocities.
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