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Additional resources for Equivalence of weak Dirichlet's principle, the method of weak solutions and Perron's method towards classical solutions of Dirichlet's problem for harmonic functions
With n = 6 and the notation of D6 and C3 as above, consider the group G = D6 × C3 × C2 . Let g1 , g2 , c ρ, be as above, and let c2 be the generator of C2 . Thehomomorphisms 1 0 0 τ and σ are defined in a similar way by ρ(g1 ) = 0 1 0 , ρ(g2 ) = 0 0 1 2π 2π −1 0 0 1 0 0 cos 3 − sin 3 0 0 −1 0 , ρ(c) = sin 2π cos 2π 0 , ρ(c2 ) = 0 1 0 , 3 3 0 0 1 0 0 −1 0 0 1 2π 2π cos 3 − sin 3 0 1 1 0 , τ(g2 ) = , τ(c) = τ(g1 ) = , τ(c2 ) = 2π 2π 1 0 0 1 sin 3 cos 3 −1 0 , σ(g1 ) = (1, 3, 2)(4, 5, 6), σ(g2 ) = (1, 4)(2, 5)(3, 6), σ(c) = 0 −1 (1, 2, 3)(4, 5, 6), σ(c2 ) = ().
Periodic solutions of singular Lagrangian systems, vol. 10 of Progress in Nonlinear Differential Equations and their Applications. Boston, MA: Birkhäuser Boston Inc. 1993 3. : Minimization properties of Hill’s orbits and applications to some N-body problems. Ann. Inst. Henri Poincar´e, Anal. Non Lin´eaire 17, 617–650 (2000) 4. : Periodic solutions of Hamiltonian systems of 3-body type. Ann. Inst. Henri Poincar´e, Anal. Non Lin´eaire 8, 561–649 (1991) 5. : Symmetries and noncollision closed orbits for planar N-bodytype problems.
9). 10) Theorem. Consider a finite group G acting on Λ so that every maximal T-isotropy subgroup of G either has the rotating circle property or acts trivially on the index set n. 8) for the n-body problem in Rd . Collisionless equivariant minimizers Proof. Let x be a local minimizer of AG in ΛG . We have two cases: either one of the T-isotropy subgroups acts trivially on n or not. 7)). 7) can be applied, and hence the minimizer x does not have interior collisions. Assume that at time t0 ∈ T a (boundary) collision occurs.