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Hetware
Posts:
148
Registered:
4/13/13


Landau & Lifschitz, Mechanics, position dependence of kinetic energy, T?
Posted:
Jan 19, 2014 12:39 PM


Please see the discussion surrounding equation (5.5) on page 9. https://ia601205.us.archive.org/11/items/Mechanics_541/LandauLifshitzMechanics.pdf
The Lagrangian is given as [Einstein summation convention assumed]
L = 1/2 a[q]_ik q'_i q'_k  U[q] (5.5)
"where the a_ik are functions of the coordinates only. The kinetic energy in generalized coordinates is still a quadratic function of the velocities, but it may depend on the coordinates also".
It's not clear what this really means. Every point q of the generalized coordinates corresponds to a point X = {x_i,y_i,z_i} in Cartesian coordinates. That means to me that U[x]=U[q[x]]. That is to say U of a given state is invariant under a change of coordinates. Since the Lagrangian is also an invariant, it seems T must be an invariant. IOW, I expect
1/2 a[q]_ik q'_i q'_k = 1/2 m_a(x_a^2 + y_a^2 + z_a^2).
Clearly the a[q]_ik are dependent on the generalized coordinates, but is the /magnitude/ of the kinetic energy coordinatedependent?



