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Study of the Second-Order Partial Differential Equation of Mass, Kinetic Energy and Force to Space-Time
DOI: 10.12677/MP.2022.126018, PDF, HTML, XML, 下载: 399  浏览: 1,140

Abstract: Derived from classical mechanics, the second-order partial differential equation of mass, kinetic energy and force with respect to time-space,  , reflects the internal relations among mass, kinetic energy and force, and their instantaneous effects in time-space. This equation is suitable for both macroscopic object and microscopic particle. Considering the relativistic effect, it is also suitable for studying the motion and force of high-speed and variable-mass object. By the equation, it is found that all kinds of basic forces have the same physical principle. The force is the result of the collision between microscopic particles. The inelastic collision with increased or tendency of increased kinetic energy will generate attraction, and other types of collision will generate repulsion. Mass is the embodiment of the change or tendency of change of kinetic energy relative to time-space. When the energy of collisions is high enough, some collisions will form mass, and others annihilate mass. This equation reveals the common nature of the four fundamental forces and the space-time principle of the interconversion of energy and mass. In particular, the research on universal gravitation and graviton has made a breakthrough.

1. 引言

2. 质量、动能与作用力关于时空的二阶偏微分方程

Figure 1. Object A moves in a straight line with uniform acceleration under the constant force F

${E}_{k}=\frac{1}{2}m{v}^{2}$ (1)

$\frac{\partial {E}_{k}}{\partial t}=mv\frac{\partial v}{\partial t}=mav$

$\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}=ma\frac{\partial v}{\partial t}=m{a}^{2}$ (2)

Ek对位移l求偏导

$\frac{\partial {E}_{k}}{\partial l}=\frac{\partial {E}_{k}}{\partial l}\frac{l}{l}=mv\frac{\partial v}{\partial l}\frac{l}{l}=m\frac{\partial l}{\partial t}\frac{\partial v}{\partial l}\frac{l}{l}=m\frac{\partial v}{\partial t}\frac{l}{l}=ma\frac{l}{l}=F\frac{l}{l}=F$ [3] (3)

${|\frac{\partial {E}_{k}}{\partial l}|}^{2}={\left(ma\right)}^{2}={m}^{2}{a}^{2}$ (4)

${|\frac{\partial {E}_{k}}{\partial l}|}^{2}=m\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (5)

1) 质能力时空方程是一个反映物体质量、动能与作用力动力学关系的二阶偏微分方程。

2) 质能力时空方程体现了物体质量、动能和作用力三者在时空方面的瞬间变化趋势的内在关系，即：质量是动能的基础，力是产生动能势的原因；反之，动能体现了质量的存在，而动能势则体现了力的存

3) 和公式F = ma相比，质能力时空方程适应范围更加广泛，特别是适应针对微观粒子的动力学分析。

4) 质能力时空方程具有普适的特点，引入相对论效应，也适应对高速变质量物体的研究。

3. 运用质能力时空方程对作用力的分析

3.1. 作用力产生的原理

$F=\frac{\partial {E}_{k}}{\partial l}=±{\left(m\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}\right)}^{1/2}\frac{l}{l}$ (6)

3.2. 作用力的斥力与引力属性分析

${F}_{1}=\frac{\partial {E}_{k1}}{\partial {l}_{1}}=-{\left(m\frac{{\partial }^{2}{E}_{k1}}{\partial {t}^{2}}\right)}^{1/2}\frac{{l}_{1}}{{l}_{1}}$ (7)

${F}_{1}=\frac{\partial {E}_{k1}}{\partial {l}_{12}}=+{\left(m\frac{{\partial }^{2}{E}_{k1}}{\partial {t}^{2}}\right)}^{1/2}\frac{{l}_{12}}{{l}_{12}}=+{\left(m\frac{{\partial }^{2}{E}_{k1}}{\partial {t}^{2}}\right)}^{1/2}\frac{-{l}_{11}}{{l}_{11}}=-{\left(m\frac{{\partial }^{2}{E}_{k1}}{\partial {t}^{2}}\right)}^{1/2}\frac{{l}_{1}}{{l}_{1}}$ (8)

Figure 2. Repulsion is generated when microparticles P1 and P2 collide elastically

${F}_{1}=\frac{\partial {E}_{k1}}{\partial {l}_{1}}=+{\left(m\frac{{\partial }^{2}{E}_{k1}}{\partial {t}^{2}}\right)}^{1/2}\frac{{l}_{1}}{{l}_{1}}$ (9)

Figure 3. Attraction is generated when particles P1 and P2 collide in elastically with increasing kinetic energy

${F}_{i}=\frac{\partial {E}_{ki}}{\partial {l}_{i}}=±{\left(m\frac{{\partial }^{2}{E}_{ki}}{\partial {t}^{2}}\right)}^{1/2}\frac{\partial {l}_{i}}{\partial {l}_{i}}$ (10)

$F=\sum {F}_{i}$ (11)

3.3. 自然界的四种基本作用力

1) 万有引力通过带质量微观粒子与引力子之间的碰撞而产生，为长程力；

2) 弱相互作用力通过强子、轻子等费米子和W、Z中间玻色子的碰撞而产生，为短程力；

3) 电磁作用力通过带电微观粒子和光子的碰撞而产生，为长程力；

4) 强相互作用力通过强子(夸克)和胶子的碰撞而产生，为短程力。

Table 1. The ratio of dimensionless coupling constants, reaction rates and relative magnitude of the four fundamental interaction forces

3.4. 万有引力

$F=G\frac{Mm}{{r}^{3}}r$ G为引力常数 [1] (12)

$\frac{{\partial }^{2}{E}_{kM}}{\partial {t}^{2}}=\frac{1}{M}{|\frac{\partial {E}_{kM}}{\partial {l}_{M}}|}^{2}=\frac{1}{M}{|F|}^{2}=\frac{{G}^{2}{m}^{2}M}{{r}^{4}}$

$\frac{{\partial }^{2}{E}_{km}}{\partial {t}^{2}}=\frac{1}{m}{|\frac{\partial {E}_{km}}{\partial {l}_{m}}|}^{2}=\frac{1}{m}{|F|}^{2}=\frac{{G}^{2}m{M}^{2}}{{r}^{4}}$

$\frac{{\partial }^{2}{E}_{kM}}{\partial {t}^{2}}/\frac{{\partial }^{2}{E}_{km}}{\partial {t}^{2}}=\frac{m}{M}$ (13)

3.5. 物质的形成与变化

4. 运用质能力时空方程对质量的分析

4.1. 质量是动能在时空中变化的体现

$m={|\frac{\partial {E}_{k}}{\partial l}|}^{2}/\frac{{\partial }^{2}{E}_{k}}{\partial {t}^{2}}$ (14)

4.2. 正反粒子碰撞

P1的质能力时空方程为：

${|\frac{\partial {E}_{k1}}{\partial {l}_{1}}|}^{2}=m\frac{{\partial }^{2}{E}_{k1}}{\partial {t}_{1}^{2}}$

P2的质能力时空方程为：

${|\frac{\partial {E}_{k2}}{\partial {l}_{2}}|}^{2}=m\frac{{\partial }^{2}{E}_{k2}}{\partial {t}_{2}^{2}}$

${|\frac{\partial {E}_{k0}}{\partial {l}_{0}}|}^{2}={m}_{0}\frac{{\partial }^{2}{E}_{k0}}{\partial {t}_{0}^{2}}$ (15)

$\frac{\partial {E}_{k0}}{\partial {l}_{0}}=\frac{\partial {E}_{k1}}{\partial {l}_{1}}+\frac{\partial {E}_{k2}}{\partial {l}_{2}}$ (16)

P1和P2互为反粒子，以相同速度相互靠近，能量相等，位移相反，所以在碰撞过程中Ek1 = Ek2l1 = l2。即

$\frac{\partial {E}_{k0}}{\partial {l}_{0}}=\frac{\partial {E}_{k1}}{\partial {l}_{1}}+\frac{\partial {E}_{k2}}{\partial {l}_{2}}=\frac{\partial {E}_{k1}}{\partial {l}_{1}}-\frac{\partial {E}_{k1}}{\partial {l}_{1}}=0$

${|\frac{\partial {E}_{k0}}{\partial {l}_{0}}|}^{2}={m}_{0}\frac{{\partial }^{2}{E}_{k0}}{\partial {t}_{0}^{2}}=0$ (17)

${E}_{k0}={E}_{k1}+{E}_{k2}$ (18)

$\frac{{\partial }^{2}{E}_{k0}}{\partial {t}_{0}^{2}}=\frac{{\partial }^{2}\left({E}_{k1}+{E}_{k2}\right)}{\partial {t}_{0}^{2}}\ne 0$ (19)

${m}_{0}=0$

4.3. 高能光子碰撞

P1和P2都为高能光子，它们发生正面碰撞，实验证明这种碰撞为弹性碰撞，它们将交换速度而分离 [1]。

P1的质能时空方程为：

${|\frac{\partial {E}_{k1}}{\partial {l}_{1}}|}^{2}={m}_{1}\frac{{\partial }^{2}{E}_{k1}}{\partial {t}_{1}^{2}}$ (20)

P2的质能时空方程为：

${|\frac{\partial {E}_{k2}}{\partial {l}_{2}}|}^{2}={m}_{2}\frac{{\partial }^{2}{E}_{k2}}{\partial {t}_{2}^{2}}$ (21)

${m}_{1}={|\frac{\partial {E}_{k1}}{\partial {l}_{1}}|}^{2}/\frac{{\partial }^{2}{E}_{k1}}{\partial {t}_{1}^{2}}$ (22)

${m}_{2}={|\frac{\partial {E}_{k2}}{\partial {l}_{2}}|}^{2}/\frac{{\partial }^{2}{E}_{k2}}{\partial {t}_{2}^{2}}$ (23)

5. 总结

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