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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.) The simplest choice might be to guess it must be proportional to v2, where v→ is the particle velocity in an inertial frame K. Take L = v2.  A second inertial frame K′ moves at constant velocity −V0→ with respect to K, so that the transformation law of velocities is v'→ = v→ + V0→.  Prove that L' = v'2 is a possible choice for the Lagrangian in the frame K′.  Explain how this proves that all inertial frames are equivalent. With this approach we prove the equivalence of inertial frames from the form of the Lagrangian, instead of postulating this equivalence at the start, which is usual way of doing things.

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.) The simplest choice might be to guess it must be proportional to v2, where v→ is the particle velocity in an inertial frame K. Take L = v2.  A second inertial frame K′ moves at constant velocity −V0→ with respect to K, so that the transformation law of velocities is v'→ = v→ + V0→.  Prove that L' = v'2 is a possible choice for the Lagrangian in the frame K′.  Explain how this proves that all inertial frames are equivalent. With this approach we prove the equivalence of inertial frames from the form of the Lagrangian, instead of postulating this equivalence at the start, which is usual way of doing things.

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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.)


The simplest choice might be to guess it must be proportional to v2,
where v→ is the particle velocity in an inertial frame K. Take L = v2.  A second inertial frame K′ moves at constant velocity −V0→ with respect to K, so that the transformation law of velocities is
v'→ = v→ + V0→.  Prove that L' = v'2 is a possible choice for the Lagrangian in the frame K′.  Explain how this proves that all inertial frames are equivalent. With this approach we prove the equivalence of inertial frames from the form of the Lagrangian, instead of postulating this equivalence at the start, which is usual way of doing things.

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