Classical Mechanics (Goldstein) · Problem 18

Survey of the Elementary Principles — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · lagrangian-point-transformation-physical-system

Problem

A Lagrangian for a particular physical system can be written as

L' = \frac{m}{2}\left(a\dot{x}^2 + 2b\dot{x}\dot{y} + c\dot{y}^2\right) - \frac{K}{2}\left(ax^2 + 2bxy + cy^2\right),

where

a

b

, and

c

are arbitrary constants but subject to the condition that

b^2 - ac \neq 0

. What are the equations of motion? Examine particularly the two cases

a = 0 = c

and

b = 0

c = -a

. What is the physical system described by the above Lagrangian? Show that the usual Lagrangian for this system as defined by Eq. (1.56) is related to

L'

by a point transformation (cf. Derivation 10). What is the significance of the condition on the value of

b^2 - ac

Related theory

The Equivalence Point And The End Point

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