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. Obtain the Euler – Lagrange equation for the extremals of the functional
X2
| [y2 - yy'+y'2 ]dlx
X1

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If L is a Lagrangian for a system of n degrees of freedom satisfying Lagrange’s equations, show by direct substitution that L' = L + (dF(q1, ..., qn, t)/dt) also satisfies Lagrange’s equations where F is any arbitrary, but differentiable, function of its arguments.
Derive using the Lagrangian and Lagrange's equation only. Please draw a diagram
Part b
Please obtain the same result as in the book.
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Consider the question of finding the points on the curve xy² = 2 closest to the origin. (a) State what function is being minimized for this problem and what the constraint is. Label each. (b) Use Lagrange multipliers to find a system of equations for finding the closest point. Write this system of equations without any vectors. Include the constraint as one of the equations. (c) Solve the system of equations from part (b) to find the points closest on the curve xy² = 2 closest to the origin.