Finding Conserved Quantities of a Given Lagrangian

[pcvrx560]

Homework Statement

Find two independent conserved quantities for a system with Lagrangian

[tex] L = A\dot{q}^{2}_{1} + B\dot{q_{1}}\dot{q_{2}} + C\dot{q}^{2}_{2} - D(2q_{1}-q_{2})^{4}\dot{q_{2}} [/tex]

where A, B, C, and D are constants.

Homework Equations

None.

The Attempt at a Solution

I've only found one symmetry,

[tex] q_{1}\rightarrow q_{1}+C [/tex]
[tex] q_{2}\rightarrow q_{2}+2C [/tex]

with the corresponding conserved quantity

[tex] 2\dot{q_{1}}(A+B) + (B+4C)\dot{q_{2}} - 2D(2\dot{q_{1}}-\dot{q_2})^{4} [/tex]

The other symmetry and conserved quantity is not so obvious to me.

This is actually a homework assignment I got back and am looking over for a test.

 

[NateD]

I understand why the first symmetry is conserved, but I don't understand why it's the only one. I feel like there should be another symmetry and corresponding conserved quantity. Can anyone lend a hand?