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In chapter 1 problem 7 of the 3rd edition of Goldstein the reader is asked to prove that the Nielsen's form of the Lagrange's equations is equivalent to Lagrange's equations.

Lagrange's equations:

$$\frac{d}{dt}\frac{\partial T}{\partial \dot{q}_j} + \frac{\partial T}{\partial q_j} = Q_j.$$

Nielsen's form:

$$\frac{\partial \dot{T}}{\partial \dot{q}_j} - 2 \frac{\partial T}{\partial q_j} = Q_j.$$

What application does this form of the equations have? It seems like Nielsen's form form would be just as difficult to calculate as Lagrange's equations. Is there a context in which Nielsen's form form is easier to use?

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No, its a form of naive algebra applied by mathematicians of the Euler age; sometimes right, sometimes obscure.

Generally, the kinetic energy is a quadratic local bilinear form on the tangent space, expressed by the vector coordinates, the velocities

$$T(\dot q, q) = \frac{1}{2} \sum g_{ik}(q) \dot q^i \dot q^k$$

and

$$\frac{d}{dt} T = \sum g_{ik}(q) \dot q^i \ddot q^k + \frac{1}{2}\sum_j\partial_j g(q) \dot q^i \dot q^k \dot q^j $$

Applying the rules of diffential algebra, a partial derivative with respect to $dot q_j$ seems to assume $\ddot q_i$ to be a constant.

So this differential rule is more of a trick to determine the coefficient in the equation, where $\ddot q$ is merely seen as another independent variable.

By the central identity of Euler-Langrange equations, the second derivatives will be quadratic forms over the velocity; so this attempt is more confusing than practicable, since the notations seems to prescribe a directional derivative on the sphere of constant velocity squared.

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