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Write down the Lagrangian for the simple pendulum of
Chapter 7, Problem 7.51(choose chapter or problem)
Write down the Lagrangian for the simple pendulum of Figure 7.2 in terms of the rectangular coordinates x and y. These coordinates are constrained to satisfy the constraint equation f y) Vx2 y2 = 1. (a) Write down the two modified Lagrange equations (7.118) and (7.119). Comparing these with the two components of Newton's second law, show that the Lagrange multiplier is (minus) the tension in the rod. Verify Equation (7.122) and the corresponding equation in y. (b) The constraint equation can be written in many different ways. For example we could have written 1(x, y) x2 + y2 = *2. t Check that using this function would have given the same physical results.
Questions & Answers
QUESTION:
Write down the Lagrangian for the simple pendulum of Figure 7.2 in terms of the rectangular coordinates x and y. These coordinates are constrained to satisfy the constraint equation f y) Vx2 y2 = 1. (a) Write down the two modified Lagrange equations (7.118) and (7.119). Comparing these with the two components of Newton's second law, show that the Lagrange multiplier is (minus) the tension in the rod. Verify Equation (7.122) and the corresponding equation in y. (b) The constraint equation can be written in many different ways. For example we could have written 1(x, y) x2 + y2 = *2. t Check that using this function would have given the same physical results.
ANSWER:Step 1 of 7
The position of the pendulum is given by x and y.
Let assume the potential energy is zero at O.
The potential energy of the pendulum is given by
Here 
The kinetic energy of the system is given by
Here, 