# hw4 - MAE 261A E NERGY AND VARIATIONAL P RINCIPLES IN S...

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MAE 261A E NERGY AND V ARIATIONAL P RINCIPLES IN S TRUCTURAL M ECHANICS F ALL 2004 Homework Assignment No. 4 Not to be collected or graded. Problem 1 Consider the minimization problem min integraldisplay b a F ( x, y, y prime ) dx + f ( z ) subject to: integraldisplay b a G ( x, y, y prime ) dx + g ( z ) = const where f ( z ) and g ( z ) are differentiable functions of z R . Recall that for the case f = g = 0 , the Euler-Lagrange equation for the constrained minimization problem is ∂F * ∂y - d dx ∂F * ∂y prime = 0 (1) where F * = F + λG and λ is a Lagrange multiplier. It can be shown that the solution to the above problem for general f and g is given by Equation (1) above plus the condition that df * dz = 0; f * = f + λg. Use this information along with the Principle of Minimum Potential Energy to solve the problem of a perfectly flexible rope hanging under the influence of gravity, as shown in Figure 1. Obtain the Euler-Lagrange equation and write down all equations necessary to determine the integration constants and the Lagrange multiplier. Determine

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• Fall '04
• Klug
• Energy, Lagrangian mechanics, Euler–Lagrange equation, Joseph Louis Lagrange

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