Drexel dragonThe Math ForumDonate to the Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math.independent

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

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

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

Posted: Jan 20, 2014 11:05 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 1/20/2014 3:11 AM, William Elliot wrote:
> On Sun, 19 Jan 2014, Hetware wrote:
>> The Lagrangian is given as [Einstein summation convention assumed]
>> L = 1/2 a[q]_ik q'_i q'_k - U[q] (5.5)

> L = sum_jk 1/2 * a_jk(q) q'_j q'_k - U(q)

'Einstein summation convention assume'

" The convention that repeated indices are implicitly summed over. This
can greatly simplify and shorten equations involving tensors. For
example, using Einstein summation,


The convention was introduced by Einstein (1916, sec. 5), who later
jested to a friend, "I have made a great discovery in mathematics; I
have suppressed the summation sign every time that the summation must be
made over an index which occurs twice..." (Kollros 1956; Pais 1982, p.
216). "

I try to reserve it's application to the contraction of contragredient
entities, but I'm being a bit lax here. Also, I am using [] to enclose
the arguments of functions. It's a convention I am trying to adapt in
order to disambiguate f(x+y) meaning 'f*x+f*y' from f(x+y) meaning 'f is
a function taking x+y as arguments'. I am not the only person who has
adapted this convention. Now, when I start talking about commutators,
I'm going to be in trouble.

>> "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

> q is the position vector; q'_j are the velocity vectors in the j-th
> 3direction.
> L(q) = (1/2)sum_jk a_jk(q) dq/dx_j dq/dx_k - U(q)
> Which for rectangular coordinate three space,
> a_jk is Kroniker's delta_jk and L = mv^2 / 2.
> since v = sqr((v_x)^2 + (v_y)^2 + (v_z)^2),
> assuming potiential energy U(q) = 0.

That part I got.

I understand that in order to calculate T in terms of generalized
coordinates, I need both q and q'.

My apprehension is regarding the notion that T=T[x'] is /not/ a function
of the Cartesian 3N-component position vector x, but T=[q,q'] /is/ a
function of /both/ the generalized /position/ and generalized /velocity/.

When I asked John Archibald Wheeler what a tensor is he replied "A
tensor is a geometric object independent of coordinates."

Since a vector is a tensor, it would seem to follow that the Cartesian
representation x should be synonymous with q.

How, then, can it be the case that T=T[q,q']=T[x']. It seems to imply
that q and q' can be expressed in terms of x' alone.

>> 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?

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.