(a) Discuss your understanding of the concepts of the symmetry of a mechanical system, a conserved quantityor quantities within the mechanical system and the relation between them. Illustrate your answer withan example, but not the example in the Lecture Notes.What is the benefit of symmetry when analysing a mechanical system?(b) Consider the Lagrangian function on R? (defined by the Cartesian coordinates (x, y)) given by1Lm (i² – ý²) + a(y² – x²),where m and a are constants.(i) Show to first order in e (that is, ignore terms of order e? and higher), that L is invariant under thetransform(x, y) + (x + €Y, Y + ex).(ii) Find the integral of motion predicted by Noether's theorem for the Lagrangian function L.

INTRODUCTION:- The existence of symmetries of Hamiltonian and Lagrangian systems is related with the…

Answered: a) Discuss your understanding of the…

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Please obtain the same result as in the book.
I need help with this problem. For 1a and 1b, I want to see how to do total differential. And for 1c, I want to see -1 in the relationship between the partial derivatives.
Consider the schematic of the single pendulum. M The kinetic energy T and potential energy V may be written as: T = ²m²²8² V = -gml cos (0) аас dt 80 The Lagrangian L is given by L=T-V, and the Euler-Lagrange equations for the motion of the pendulum are given by the following second order differential equation in : ас 80 = 11 = 0 Write down the second order ODE using the specific T and V defined above. Please write this ODE in the form = f(0,0). Notice that this ODE is not linear! Now you may assume that l = m = g = 1 for the remainder of the problem. You may still suspend variables to get a system of two first order (nonlinear) ODEs by writing the ODE as: w = f(0,w) What are the fixed points of this system where all derivatives are zero? Write down the linearized equations in a neighborhood of each fixed point and determine the linear stability. You may formally linearize the nonlinear ODE or you may use a small angle approximation for sin(0); the two approaches are equivalent.
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.