- #1
motion_ar
- 36
- 1
The principle of least action applicable in an uniform field can be obtained as follows:
Particle A
[tex]\vec{a}_A = \vec{a}_A[/tex]
[tex]\int \vec{a}_A \cdot d\vec{r}_A = \int \vec{a}_A \cdot d\vec{r}_A[/tex]
[tex]\int \vec{a}_A \cdot d\vec{r}_A = \Delta \; {\textstyle \frac{1}{2}}\vec{v}_A^2[/tex]
[tex]\int \vec{a}_A \cdot d\vec{r}_A = \Delta \; \; \vec{a}_A \cdot \vec{r}_A[/tex]
[tex]\Delta \; {\textstyle \frac{1}{2}}\vec{v}_A^2 = \Delta \; \; \vec{a}_A \cdot \vec{r}_A[/tex]
[tex]\Delta \; {\textstyle \frac{1}{2}}\vec{v}_A^2 - \Delta \; \; \vec{a}_A \cdot \vec{r}_A = 0[/tex]
[tex] m_A \left( \Delta \; {\textstyle \frac{1}{2}}\vec{v}_A^2 - \Delta \; \; \vec{a}_A \cdot \vec{r}_A \right) = 0[/tex]
Alex
Particle A
[tex]\vec{a}_A = \vec{a}_A[/tex]
[tex]\int \vec{a}_A \cdot d\vec{r}_A = \int \vec{a}_A \cdot d\vec{r}_A[/tex]
[tex]\int \vec{a}_A \cdot d\vec{r}_A = \Delta \; {\textstyle \frac{1}{2}}\vec{v}_A^2[/tex]
[tex]\int \vec{a}_A \cdot d\vec{r}_A = \Delta \; \; \vec{a}_A \cdot \vec{r}_A[/tex]
[tex]\Delta \; {\textstyle \frac{1}{2}}\vec{v}_A^2 = \Delta \; \; \vec{a}_A \cdot \vec{r}_A[/tex]
[tex]\Delta \; {\textstyle \frac{1}{2}}\vec{v}_A^2 - \Delta \; \; \vec{a}_A \cdot \vec{r}_A = 0[/tex]
[tex] m_A \left( \Delta \; {\textstyle \frac{1}{2}}\vec{v}_A^2 - \Delta \; \; \vec{a}_A \cdot \vec{r}_A \right) = 0[/tex]
Alex