Conserved Quantity

Classical Mechanics Level 3

A physical system in three spatial coordinates $(x,y,z)$ is described by the following Lagrangian :

$\large{L = x^2 \dot{y} + 2 z^3 \dot{x} \, \dot{y}^2 + 3 x \dot{x} \dot{z}}$

Which quantity must be conserved?

Details and Assumptions:
- The dot convention denotes time differentiation

\large{2 x \dot{y} + 3 \dot{x} \dot{z}}

\large{2 z^3 \dot{y}^2 +3 x \dot{z}}

\large{6 z^2 \dot{x} \dot{y}^2}

\large{x^2 + 4 z^3 \dot{x} \dot{y} }

\large{3 x \dot{x}}

1 solution

Steven Chase
Jul 29, 2017

Consider the Euler-Lagrange equation for the $y$ variable (similar equations exist for the other two as well):

$\large{\frac{d}{dt} \frac{\partial{L}}{\partial \dot{y}} = \frac{\partial{L}}{\partial y}}$

Inspecting the Lagrangian, we observe that the variable $y$ does not appear, but the other two do. Therefore, the Euler-Lagrange equation for $y$ reduces to:

$\large{\frac{d}{dt} \frac{\partial{L}}{\partial \dot{y}} = 0}$

Obviously then, the time derivative of $\Large{\frac{\partial{L}}{d \dot{y}}}$ is zero, meaning that this quantity is conserved.

$\large{\frac{\partial{L}}{d \dot{y}} = x^2 + 4z^3 \dot{x} \dot{y}}$

Conserved Quantity

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