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Let be a function
,
, and
. Begin with the time-relative Euler-Lagrange condition
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(1) |
If
, then the Euler-Lagrange condition reduces to
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(2) |
which is the Beltrami identity. In the calculus of variations, the ability to use the Beltrami identity can vastly simplify problems, and as it happens, many physical problems have
.
In space-relative terms, with
, we have
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(3) |
If
, then the Euler-Lagrange condition reduces to
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(4) |
To derive the Beltrami identity, note that
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(5) |
Multiplying (1) by , we have
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(6) |
Now, rearranging (5) and substituting in for the rightmost term of (6), we obtain
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(7) |
Now consider the total derivative
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(8) |
If
, then we can substitute in the left-hand side of (8) for the leading portion of (7) to get
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(9) |
Integrating with respect to , we arrive at
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(10) |
which is the Beltrami identity.
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