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Geometrical approach to the dynamics of a relativistic string
B. M. Barbashov,
A. L. Koshkarov
Abstract:
The problems of the classical dynamics of a relativistic string are intimately related to the theory of two-dimensional extremal surfaces in
$n$-dimensional pseudo-Euclidean space
$E^1_n$. In three-dimensional space-time
$E^1_3$, it is possible to exploit fully the formalism of the Gaussian theory of two-dimensional surfaces, the surface being specified to within shifts by its first and second quadratic forms. Integration of the derivation formulas for the basic vectors
$\partial x_\mu(\tau,\sigma)/\partial\tau=\dot x_\mu(\tau,\sigma)$,
$\partial x_\mu(\tau,\sigma)/\partial\sigma=x_\mu'(\tau,\sigma)$
are the tangent vectors to the surface and
$m_\mu(\tau,\sigma)$ is the normal to the surface at the given point
$\tau,\sigma$) yields a representation for
these vectors in a natural basis satisfying the orthonormal gauge
$(\dot x_\mu\pm x'_\mu)^2=0$ and d'Alembert's equation
$\ddot x_\mu(\tau,\sigma)-x''_\mu(\tau,\sigma)=0$ in the string dynamics. This representation can be generalized to a pseudo-Euclidean space
$E^1_n$, of any dimension
$n$. For a relativistic string in
$E^1_n$ a representation is obtained that contains
$n-2$ arbitrary functions and satisfies the gauge conditions, the equations of motion, and the boundary conditions for a free string.
Received: 14.04.1978
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