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Doubled variables appear in specific applications as early as [ 16 ] and more recently in [ 17 , 18 ] . However, this paper constructs a general framework relating the doubled variables, Hamilton’s principle, initial conditions, and non-conservative dynamics for general dynamical systems

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The same is true in the general case if K = V 1 ​ ( \mathaccentV ​ v ​ e ​ c ​ 17 ​ E ​ q 1 , \mathaccentV ​ d ​ o ​ t ​ 05 ​ F ​ \mathaccentV ​ v ​ e ​ c ​ 17 ​ E ​ q 1 ) − V 2 ​ ( \mathaccentV ​ v ​ e ​ c ​ 17 ​ E ​ q 2 , \mathaccentV ​ d ​ o ​ t ​ 05 ​ F ​ \mathaccentV ​ v ​ e ​ c ​ 17 ​ E ​ q 2 ) K=V_{1}({\mathaccentV{vec}17E{q}}_{1},\mathaccentV{dot}05F{{\mathaccentV{vec}17E{q}}}_{1})-V_{2}({\mathaccentV{vec}17E{q}}_{2},\mathaccentV{dot}05F{{\mathaccentV{vec}17E{q}}}_{2}) for V 1 ≠ V 2 V_{1}\not=V_{2} . The resulting equations of motion are the same as if ( V 1 + V 2 ) / 2 (V_{1}+V_{2})/2 is absorbed into each Lagrangian

Cited in the paper.

Using Legendre transforms with respect to \mathaccentV ​ v ​ e ​ c ​ 17 ​ E ​ p a {\mathaccentV{vec}17E{p}}_{a} or \mathaccentV ​ v ​ e ​ c ​ 17 ​ E ​ π a {\mathaccentV{vec}17E{\pi}}_{a} leads to different but related Hamiltonian formulations that will be detailed elsewhere

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