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While discussing the small oscillations of particles about a stable equilibrium, Landau writes

Where q is the generalized co-ordinate.

Section 21, Volume 1

1. How do you know such a polynomial expansion for q is allowed? How do you know it exists? After all, this is any old U with its first derivative 0 at q

2. Why does the co-efficient of [tex]\dot{q}^{2}[/tex] have to be a function of q? I thought it'd be a constant.

...The potential U(q) for small deviations can be expressed as a polynomial

[tex] U(q) - U(q_{0}) = \frac{1}{2}k(q - q_{0})^{2} [/tex]

...

The kinetic energy of a free particle in one dimension is generally of the form

[tex]\frac{1}{2}a(q)\dot{q}^{2}[/tex]

...

Where q is the generalized co-ordinate.

Section 21, Volume 1

1. How do you know such a polynomial expansion for q is allowed? How do you know it exists? After all, this is any old U with its first derivative 0 at q

_{0}.2. Why does the co-efficient of [tex]\dot{q}^{2}[/tex] have to be a function of q? I thought it'd be a constant.

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