Verify whether the following candidate functions are first…
Verify whether the following candidate functions are first integrals of the symmetric system: $$ \frac{dx}{x} = \frac{dy}{-y} = \frac{dz}{z};\qquad \text{test } \Phi_1=\frac{x+y}{z+x},\ \ \Phi_2=\frac{z-y}{x+y} $$
Explanation of Method
Rather than solving the system from scratch, differentiate each candidate quantity along the flow $\dot x=x,\ \dot y=-y,\ \dot z=z$ directly; a first integral must have zero total derivative, and any nonzero remainder disqualifies it, pointing instead to the correct conserved quantities.
- Write the system in explicit form $\dot x=x,\ \dot y=-y,\ \dot z=z$.
- Differentiate $\Phi_1$ along the flow and check whether it vanishes.
- Differentiate $\Phi_2$ along the flow and check whether it vanishes.
- If both fail, derive the correct first integrals directly from the ratios.
Step 1 · Test $\Phi_1=(x+y)/(z+x)$
Using $\dot x+\dot y=x-y$ and $\dot z+\dot x=z+x$,
$$ \frac{d\Phi_1}{d\tau}=\frac{(x-y)(z+x)-(x+y)(z+x)}{(z+x)^2}=\frac{-2y}{z+x}\neq0. $$
So $\Phi_1$ is not a first integral.
Step 2 · Test $\Phi_2=(z-y)/(x+y)$
Using $\dot z-\dot y=z+y$ and $\dot x+\dot y=x-y$,
$$ \frac{d\Phi_2}{d\tau}=\frac{(z+y)(x+y)-(z-y)(x-y)}{(x+y)^2}=\frac{2y(z+x)}{(x+y)^2}\neq0. $$
So $\Phi_2$ is also not a first integral.
Step 3 · Find the correct first integrals
Directly from the ratios: $\dfrac{dx}{x}=-\dfrac{dy}{y}$ integrates to $xy=\text{const}$, and $\dfrac{dx}{x}=\dfrac{dz}{z}$ integrates to $x/z=\text{const}$:
$$ \Psi_1=xy=C_1,\qquad \Psi_2=\frac{x}{z}=C_2. $$
Step 4 · Verification
Differentiating, $\dfrac{d(xy)}{d\tau}=xy+x(-y)=0$ and $\dfrac{d(x/z)}{d\tau}=(xz-xz)/z^2=0$, confirming $\Psi_1$ and $\Psi_2$ are genuine first integrals. ✓
Summary Table
| Quantity | Value |
|---|---|
| $\Phi_1=(x+y)/(z+x)$ | not conserved: $\dot\Phi_1=-2y/(z+x)$ |
| $\Phi_2=(z-y)/(x+y)$ | not conserved: $\dot\Phi_2=2y(z+x)/(x+y)^2$ |
| Correct integrals | $\Psi_1=xy=C_1,\ \Psi_2=x/z=C_2$ |
Additional Notes
- This problem is a caution against pattern-matching: $\Phi_1$ and $\Phi_2$ look plausible (they are functionally independent, as a Jacobian check confirms), but neither is actually constant along the flow.
- The only reliable test for a first integral is direct differentiation along the system, not visual resemblance to known conserved forms.
- The genuine first integrals $xy$ and $x/z$ follow immediately from pairing ratios with matching (or opposite) signs.
Result
The correct, verified first integrals of the system are: