Before we get back to geometry let's have some fun with alternate ways to do classical mechanics!

– I mentioned Lagrangian and Hamiltonian mechanics earlier. Lagrange's is often simpler, but it cannot handle cyclic coordinates. Meet Routhian mechanics! Routh found out that you can cherry-pick momenta or velocities as your generalized coordinates to your delight.

en.wikipedia.org/wiki/Routhian

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> – Appell's equation of motion […] is an alternative general formulation of classical mechanics described by Paul Émile Appell in 1900 and Josiah Willard Gibbs in 1879.

Gibbs! Who invented vector calculus, and coined the term "statistical mechanics"!

en.wikipedia.org/wiki/Appell%2

It uses the second derivative to make things solveable. This approach shines when nonholonomic constraints are involved.

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> – The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities.

en.wikipedia.org/wiki/Hamilton

> In which the motion of a particle can be represented as a wave.

> The HJE is a single, first-order partial differential equation for the function S of the N generalized coordinates q1...qN and the time t. The generalized momenta do not appear, except as derivatives of S.

Just like Lagrangian mechanics! But the latter amounts to a system of N equations.

– The Udwadia–Kalaba equation is useful when forces aren't conservative (they don't obey d'Alembert's principle).

en.wikipedia.org/wiki/Udwadia%

M(q,t)q''(t) = Q(q,q',t)

Q is the total (generalized) force, M the mass matrix. M has to be symmetric, and semi-positive definite.

No Lagrange multipliers! It is based on Gauss' principle instead of Euler-Lagrange's equation.

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