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Topic: Landau & Lifschitz, Mechanics, position dependence of kinetic energy,

Replies: 3   Last Post: Jan 20, 2014 10:15 PM

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Posts: 148
Registered: 4/13/13
Landau & Lifschitz, Mechanics, position dependence of kinetic energy,

Posted: Jan 19, 2014 12:39 PM
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Please see the discussion surrounding equation (5.5) on page 9.

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 coordinate-dependent?

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