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What You Got Right

• Correct strategy

  The student correctly recognizes that the proof amounts to showing that dF/dt contributes equally to both sides of the Euler-Lagrange equation, so its net contribution vanishes. This is exactly the right approach.

• Both key terms computed

  The student correctly computes both the partial-derivative-with-respect-to-q term and the velocity term, showing both equal d/dt(∂F/∂q_i), which is the crux of the proof.

• Correct use of the total time derivative operator

  Expanding dF/dt = Σ_j (∂F/∂q_j) q̇_j is correct since F depends only on the coordinates, and the use of Schwarz's theorem to swap partial derivatives is valid.

• Clean use of the Kronecker delta

  The evaluation ∂q̇_j/∂q̇_i = δ_{ij} to collapse the sum in the velocity term is done correctly and cleanly.

Improvements

• Circular/backwards logic in opening framing

  The student begins by asserting that L' satisfies the Euler-Lagrange equations (writing ∂L'/∂q_i = d/dt(∂L'/∂q̇_i)) as if it were already known. The logical structure of the proof should be to compute the Euler-Lagrange expression for L' and show it equals the expression for L, not to assume L' satisfies the equations and then derive that L does.

  Suggestion: Define the Euler-Lagrange operator EL_i(·) = d/dt(∂·/∂q̇_i) - ∂·/∂q_i. Show by linearity that EL_i(L') = EL_i(L) + EL_i(dF/dt), then prove EL_i(dF/dt) = 0 using exactly the two computations you already did.

• Notational inconsistencies in the total derivative operator

  The expression d/dt = Σ_j q̇_j ∂/∂q_j + ∂/∂q̇_j + ∂/∂t is written with the sum applied inconsistently — the ∂/∂q̇_j term should carry a q̈_j coefficient (and is inside the sum), and ∂/∂t should sit outside the sum. Though harmless here since F has no velocity or explicit time dependence, the operator as written is not correct.

  Suggestion: Write d/dt = Σ_j (q̇_j ∂/∂q_j + q̈_j ∂/∂q̇_j) + ∂/∂t. Then note F depends only on coordinates so the q̈ and ∂/∂t terms drop, giving d/dt = Σ_j q̇_j ∂/∂q_j acting on F or its derivatives.

• Missing explicit statement of the conclusion

  The student concludes the equations of motion are identical, which is correct, but could more explicitly state that because the extra term contributes zero to the Euler-Lagrange expression, setting EL_i(L') = 0 yields exactly EL_i(L) = 0.