By Richard A. Matzner

A sophisticated physics textbook that explains the mathematical and actual innovations of mechanics and their courting to different branches of physics. themes coated contain tensor research, variational ideas and Lagrangians, canonical differences and estimation thoughts.

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The endpoints fix q0 and q1 , but the stationary condition gives an infinite number of equations ∂I/∂an = 0. It is not really necessary to be so rigorous, however. Under a change q(t) → q(t) + δq(t), the derivative will vary by δ q˙ = d δq(t)/dt, and the 48 CHAPTER 2. LAGRANGE’S AND HAMILTON’S EQUATIONS functional I will vary by δI = ∂f ∂f δq + δ q˙ dt ∂q ∂ q˙ f = ∂f δq + ∂ q˙ i ∂f d ∂f − δqdt, ∂q dt ∂ q˙ where we integrated the second term by parts. 17) We see that if f is the Lagrangian, we get exactly Lagrange’s equation.

It is not really necessary to be so rigorous, however. Under a change q(t) → q(t) + δq(t), the derivative will vary by δ q˙ = d δq(t)/dt, and the 48 CHAPTER 2. LAGRANGE’S AND HAMILTON’S EQUATIONS functional I will vary by δI = ∂f ∂f δq + δ q˙ dt ∂q ∂ q˙ f = ∂f δq + ∂ q˙ i ∂f d ∂f − δqdt, ∂q dt ∂ q˙ where we integrated the second term by parts. 17) We see that if f is the Lagrangian, we get exactly Lagrange’s equation. The above derivation is essentially unaltered if we have many degrees of freedom qi instead of just one.

DL = dt i ∂L dqi + ∂qi dt i ∂L dq˙i ∂L + ∂ q˙i dt ∂t In the first term the first factor is d ∂L dt ∂ q˙i by the equations of motion, so dL d = dt dt i ∂L ∂L q˙i + . e. when ∂L/∂t = 0, so we rewrite this in terms of q˙i H(q, q, ˙ t) = i 6 ∂L −L= ∂ q˙i q˙i Pi − L i It seems curious that we are finding straightforwardly one of the components of the conserved momentum, but not the other two, Ly and Lx , which are also conserved. The fact that not all of these emerge as conjugates to ignorable coordinates is related to the fact that the components of the angular momentum do not commute in quantum mechanics.