Lagrange + Friction

Classical Mechanics Level 2

Suppose we have a block of mass m m sliding on a rough surface with a uniform kinetic friction coefficient μ \mu . The block initially begins with some speed at position x = 0 x = 0 and slides in the positive x x direction. The downward gravitational acceleration is g g .

Friction is a non-conservative force, so it is ordinarily not convenient to use Lagrange analysis in such a case. However, in the interval between the block beginning its slide and the block coming to rest, we can use Lagrange analysis by modeling the energy lost to friction as a potential. The analysis ceases to be valid after the block stops.

During the period over which the Lagrange model is valid, what is the effective Lagrangian?

1 2 m x ˙ 2 + m g x \frac{1}{2} m \, \dot{x}^2 + m g \, x 1 2 m x ˙ 2 + μ m g x \frac{1}{2} m \, \dot{x}^2 + \mu \, m g \, x 1 2 m x ˙ 2 − m g x \frac{1}{2} m \, \dot{x}^2 - m g \, x 1 2 m x ˙ 2 − μ m g x \frac{1}{2} m \, \dot{x}^2 - \mu \, m g \, x

Steven Chase by Steven Chase , 37, USA

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1 solution

We have to consider the work done by the non-conservative force as potential (effective) of the block such that it will account for the lost energy of the block.

U = F . x = μ m g x U = F.x = \mu mgx .

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