Beltrami identity

Let q(t) be a function , q˙=ddtq, and L=L(q,q˙,t). Begin with the time-relative Euler-Lagrange conditionPlanetmathPlanetmath

qL-ddt(q˙L)=0. (1)

If tL=0, then the Euler-Lagrange condition reduces to

L-q˙q˙L=C, (2)

which is the Beltrami identityMathworldPlanetmath. In the calculus of variationsMathworldPlanetmath, the ability to use the Beltrami identity can vastly simplify problems, and as it happens, many physical problems have tL=0.

In space-relative terms, with q:=ddxq, we have

qL-ddxqL=0. (3)

If xL=0, then the Euler-Lagrange condition reduces to

L-qqL=C. (4)

To derive the Beltrami identity, note that

ddt(q˙q˙L)=q¨q˙L+q˙ddt(q˙L) (5)

Multiplying (1) by q˙, we have

q˙qL-q˙ddt(q˙L)=0. (6)

Now, rearranging (5) and substituting in for the rightmost term of (6), we obtain

q˙qL+q¨q˙L-ddt(q˙q˙L)=0. (7)

Now consider the total derivative

ddtL(q,q˙,t)=q˙qL+q¨q˙L+tL. (8)

If tL=0, then we can substitute in the left-hand side of (8) for the leading portion of (7) to get

ddtL-ddt(q˙q˙L)=0. (9)

Integrating with respect to t, we arrive at

L-q˙q˙L=C, (10)

which is the Beltrami identity.

Title Beltrami identity
Canonical name BeltramiIdentity
Date of creation 2013-03-22 12:21:08
Last modified on 2013-03-22 12:21:08
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 8
Author PrimeFan (13766)
Entry type Definition
Classification msc 47A60
Related topic CalculusOfVariations
Related topic EulerLagrangeDifferentialEquation