Assume that you do not know about the kinetic energy or Newton’s Laws of motion. Suppose instead of deriving the Euler-Lagrange equations, we postulate them. We define the basic law of mechanics to be these equations and ask ourselves the question: What is the Lagrangian for a free particle? (This is a particle in empty space with no forces acting on it. Be sure to set up an inertial reference system.) Explain why, on very general grounds, L can not be a function of x, y, and z. It also can not depend on the individual coordinates of velocity in any way except as a function of the magnitude of the velocity: v2 = vx2 + vy2 + vz2. On what assumption about the properties of space does this depend?

Assume that you do not know about the kinetic energy or Newton’s Laws of motion. Suppose instead of deriving the Euler-Lagrange equations, we postulate them. We define the basic law of mechanics to be these equations and ask ourselves the question: What is the Lagrangian for a free particle? (This is a particle in empty space with no forces acting on it. Be sure to set up an inertial reference system.) Explain why, on very general grounds, L can not be a function of x, y, and z. It also can not depend on the individual coordinates of velocity in any way except as a function of the magnitude of the velocity: v2 = vx2 + vy2 + vz2. On what assumption about the properties of space does this depend?

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Question

Explain why, on very general grounds, L can not be a function of x, y,
and z. It also can not depend on the individual coordinates of velocity in any way except as a function of the magnitude of the velocity: v² = v_x²+ v_y²+ v_z². On what assumption about the properties of space does this depend?

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