Manuscript II@2 · II.tex · section · 50d64def

The exact contact evaluator

Locked kinematics, action, and contacts › The exact contact evaluator · lines 618–654

II@2#eq:Contr, II@2#eq:contact-evaluator

For lower-index momenta $p, q$ , define

Contr_{P} (p, q; e_{1}, e_{2}, e_{3}, e_{4}) = λ_{1}, \dots, λ_{10} = 0 \sum^{3} p_{λ_{1}} q_{λ_{2}} r = 1 \prod^{4} (e_{r})_{λ_{2 r + 1} λ_{2 r + 2}} \times {u, v} \in P \prod η^{λ_{u} λ_{v}} .

The differentiated response of a raw contact is

n_{I, P} (β) = - σ \in S_{4} \sum (r = 1 \prod^{4} δ_{I_{r}, J_{σ (r)}}) \times Contr_{P} (k_{σ (1)}, k_{σ (2)}; e_{σ (1)}, e_{σ (2)}, e_{σ (3)}, e_{σ (4)}) .

One $e_{i}$ is the prescribed $G$ . The momenta in $Contr_{P}$ have lower indices.

To derive this formula, substitute

h^{I} = j \sum z_{j} δ_{I, J_{j}} e_{j} e^{i k_{j} \cdot x}

in (eq:raw) and extract $z_{1} z_{2} z_{3} z_{4}$ . All assignments to the factor positions appear. The two derivatives contribute $i^{2} = - 1$ .

Displayed formulas in this passage

II@2#eq:Contr · line 621
$Contr_{P} (p, q; e_{1}, e_{2}, e_{3}, e_{4}) = λ_{1}, \dots, λ_{10} = 0 \sum^{3} p_{λ_{1}} q_{λ_{2}} r = 1 \prod^{4} (e_{r})_{λ_{2 r + 1} λ_{2 r + 2}} \times {u, v} \in P \prod η^{λ_{u} λ_{v}} .$
II@2#eq:contact-evaluator · line 632
$n_{I, P} (β) = - σ \in S_{4} \sum (r = 1 \prod^{4} δ_{I_{r}, J_{σ (r)}}) \times Contr_{P} (k_{σ (1)}, k_{σ (2)}; e_{σ (1)}, e_{σ (2)}, e_{σ (3)}, e_{σ (4)}) .$
unlabeled · line 648
$h^{I} = j \sum z_{j} δ_{I, J_{j}} e_{j} e^{i k_{j} \cdot x}$