數學物理是數學和物理學的交叉領域,指應用特定的數學方法來研究物理學的某些部分。對應的數學方法也叫數學物理方法。數學和物理學的發展在歷史上一直密不可分,許多數學理論是在物理問題的基礎上發展起來的;很多數學方法和工具通常也只在物理學中找到實際應用。不過,也只是互相參考而已,所有沒有所謂的一定。[1]
主要內容
- 微分方程式的解算:很多物理問題,比如在古典力學和量子力學中求解運動方程式,都可以被歸結為在一定邊界條件下的對微分方程式的求解。因此求解微分方程式成為數學物理的最重要組成部分。相關的數學工具包括:
- 場的研究(場論):場是現代物理的主要研究對象。電動力學研究電磁場;廣義相對論研究重力場;規範場論研究規範場。對不同的可使用不同的數學工具,包括:
- 對稱性的研究:對稱性是物理中的重要概念。它是守恆律的基礎,在晶體學和量子場論中都有重要應用。對稱性由對稱群或相關的代數結構描述,研究它的數學工具是:
- 作用量(action)理論:作用量理論被廣泛應用於物理學的各個領域,例如分析力學和路徑積分。相關的數學工具包括:
參見
參考
文獻
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