{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-harmonic-oscillator-model"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-harmonic-oscillator-model-01","title":{"zh-Hant":"線性簡諧振子模型：資格、k/m、多項式候選的殘差與單點能量","en":"Linear harmonic oscillator: eligibility, k/m, residuals of a polynomial candidate and single-state energy"},"summary":{"zh-Hant":"取 m=2、k=8。核對模型資格並算出 k/m；對多項式候選 x_c(t)=1−2t² 算出殘差 R(t)=m x_c''+k x_c，在指定點 t=0、1/2、1 求值，並證明 x_c 不是任何正長度區間上的解；證明方程 2x''+8x=0 沒有非零多項式解；最後算出單一狀態 (x,v)=(1,0) 的能量 E=4，並以能量導數恆等式說明單點能量值為何不證明守恆。","en":"Take m=2 and k=8. Check that the parameters satisfy the model's requirements and compute k/m; for the polynomial candidate x_c(t)=1−2t², compute the residual R(t)=m x_c''+k x_c, evaluate it at the specified points t=0, 1/2, 1, and prove that x_c is not a solution on any interval of positive length; prove that the equation 2x''+8x=0 has no nonzero polynomial solution; finally compute the energy E=4 of the single state (x,v)=(1,0), and use the energy-derivative identity to explain why a single energy value proves no conservation."},"mko_ids":["mko-harmonic-oscillator-model"]}],"title":"Concept derivation guides"}