数学物理是数学和物理学的交叉领域,指应用特定的数学方法来研究物理学的某些部分。对应的数学方法也叫数学物理方法。数学和物理学的发展在历史上一直密不可分,许多数学理论是在物理问题的基础上发展起来的;很多数学方法和工具通常也只在物理学中找到实际应用。不过,也只是互相参考而已,所有没有所谓的一定。[1]
主要内容
- 微分方程的解算:很多物理问题,比如在经典力学和量子力学中求解运动方程,都可以被归结为在一定边界条件下的对微分方程的求解。因此求解微分方程成为数学物理的最重要组成部分。相关的数学工具包括:
- 场的研究(场论):场是现代物理的主要研究对象。电动力学研究电磁场;广义相对论研究引力场;规范场论研究规范场。对不同的可使用不同的数学工具,包括:
- 对称性的研究:对称性是物理中的重要概念。它是守恒律的基础,在晶体学和量子场论中都有重要应用。对称性由对称群或相关的代数结构描述,研究它的数学工具是:
- 作用量(action)理论:作用量理论被广泛应用于物理学的各个领域,例如分析力学和路径积分。相关的数学工具包括:
参见
参考
文献
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