Formula Binet (mechanics)

The Binet formula (mechanics) is a partial differential equation that allows one to determine the central force if the equation of the trajectory of a material point moving under its action is known, or to determine the trajectory by a given central force.

Content

Wording

Let the material point with mass $m$ ${\ displaystyle m}$ moves under the influence of a central force ${\vec {F}}$ ${\ displaystyle {\ vec {F}}}$ . Then in the polar coordinate system $r$ ${\ displaystyle r}$ , $\varphi$ ${\ displaystyle \ varphi}$

{\frac {d^{2}}{d\varphi ^{2}}}\left({\frac {1}{r}}\right)+{\frac {1}{r}}=-{\frac {F_{r}}{mc^{2}}}r^{2}

{\ displaystyle {\ frac {d ^ {2}} {d \ varphi ^ {2}}} \ left ({\ frac {1} {r}} \ right) + {\ frac {1} {r}} = - {\ frac {F_ {r}} {mc ^ {2}}} r ^ {2}}

.

Here $c$ ${\ displaystyle c}$ - the so-called constant squares.

Conclusion

Consider the motion of a material point. $m$ ${\ displaystyle m}$ under the influence of the central force ${\vec {F}}$ ${\ displaystyle {\ vec {F}}}$ . Equation of motion of a point $m{\vec {w}}={\vec {F}}$ ${\ displaystyle m {\ vec {w}} = {\ vec {F}}}$ in projections on the polar axis $mw_{r}=F_{r}$ ${\ displaystyle mw_ {r} = F_ {r}}$ , $mw_{\varphi }=F_{\varphi }$ ${\ displaystyle mw _ {\ varphi} = F _ {\ varphi}}$ where $F_{\varphi }=0$ ${\ displaystyle F _ {\ varphi} = 0}$ . Radial acceleration $w_{r}={\ddot {r}}-r{\dot {\varphi }}^{2}$ ${\ displaystyle w_ {r} = {\ ddot {r}} - r {\ dot {\ varphi}} ^ {2}}$ transverse acceleration $w_{\varphi }={\ddot {\varphi }}r+2{\dot {r}}{\dot {\varphi }}$ ${\ displaystyle w _ {\ varphi} = {\ ddot {\ varphi}} r + 2 {\ dot {r}} {\ dot {\ varphi}}}$ . Get $m({\ddot {r}}-r{\dot {\varphi }}^{2})=F_{r}$ ${\ displaystyle m ({\ ddot {r}} - r {\ dot {\ varphi}} ^ {2}) = F_ {r}}$ , $m({\ddot {\varphi }}r+2{\dot {r}}{\dot {\varphi }})=0$ ${\ displaystyle m ({\ ddot {\ varphi}} r + 2 {\ dot {r}} {\ dot {\ varphi}}) = 0}$ . Transform the second equation: $m({\ddot {\varphi }}r+2{\dot {r}}{\dot {\varphi }})={\frac {1}{r}}{\frac {d}{dt}}(r^{2}{\dot {\varphi }})=0$ ${\ displaystyle m ({\ ddot {\ varphi}} r + 2 {\ dot {r}} {\ dot {\ varphi}}) = {\ frac {1} {r}} {\ frac {d} { dt}} (r ^ {2} {\ dot {\ varphi}}) = 0}$ Consequently: $r^{2}{\dot {\varphi }}=c$ ${\ displaystyle r ^ {2} {\ dot {\ varphi}} = c}$ where $c$ ${\ displaystyle c}$ - constant, called constant squares. Substituting the value of \ dot \ varphi from $r^{2}{\dot {\varphi }}=c$ ${\ displaystyle r ^ {2} {\ dot {\ varphi}} = c}$ into the equation $m({\ddot {r}}-r{\dot {\varphi }}^{2})=F_{r}$ ${\ displaystyle m ({\ ddot {r}} - r {\ dot {\ varphi}} ^ {2}) = F_ {r}}$ get $m({\ddot {r}}-{\frac {c^{2}}{r^{3}}})=F_{r}$ ${\ displaystyle m ({\ ddot {r}} - {\ frac {c ^ {2}} {r ^ {3}}}) = F_ {r}}$ . Consistently find ${\dot {r}}={\frac {dr}{d\varphi }}{\dot {\varphi }}={\frac {c}{r^{2}}}{\frac {dr}{d\varphi }}=-c{\frac {d}{d\varphi }}({\frac {1}{r}})$ ${\ displaystyle {\ dot {r}} = {\ frac {dr} {d \ varphi}} {\ dot {\ varphi}} = {\ frac {c} {r ^ {2}}} {\ frac { dr} {d \ varphi}} = - c {\ frac {d} {d \ varphi}} ({\ frac {1} {r}})}$ , ${\ddot {r}}={\frac {d{\dot {r}}}{dt}}={\frac {d{\dot {r}}}{d\varphi }}{\dot {\varphi }}=-{\frac {c^{2}}{r^{2}}}{\frac {d^{2}}{d\varphi ^{2}}}({\frac {1}{r}})$ ${\ displaystyle {\ ddot {r}} = {\ frac {d {\ dot {r}}} {dt}} = {\ frac {d {\ dot {r}}} {d \ varphi}} {\ dot {\ varphi}} = - {\ frac {c ^ {2}} {r ^ {2}}} {\ frac {d ^ {2}} {d \ varphi ^ {2}}} ({\ frac {1} {r}})}$ . Substituting ${\ddot {r}}$ ${\ displaystyle {\ ddot {r}}}$ at $m({\ddot {r}}-{\frac {c^{2}}{r^{3}}})=F_{r}$ ${\ displaystyle m ({\ ddot {r}} - {\ frac {c ^ {2}} {r ^ {3}}}) = F_ {r}}$ find ${\frac {d^{2}}{d\varphi ^{2}}}({\frac {1}{r}})+{\frac {1}{r}}=-{\frac {F_{r}}{mc^{2}}}r^{2}$ ${\ displaystyle {\ frac {d ^ {2}} {d \ varphi ^ {2}}} ({\ frac {1} {r}}) + {\ frac {1} {r}} = - {\ frac {F_ {r}} {mc ^ {2}}} r ^ {2}}$ .

Literature

Moses Iosifovich Bat. Theoretical mechanics in examples and problems: Dynamics . - Science, 1968. - p. 14. - 632 p.
Mikhail M. Gernet. The course of theoretical mechanics . - Higher school, 1973. - S. 325. - 461 p.
Zhukovsky N. Ye. Analytical mechanics . - Imperial Moscow Technical School , 1910. - 262 p. - ISBN 9785446096053 .

Notes

↑ Bugaenko G. A. , Malanin V. V. , Yakovlev V. I. Basics of Classical Mechanics. - M.: Higher School, 1999. - S. 86-87. - ISBN 5-06-003587-5

[Бугаенко-1] Bugaenko G. A. , Malanin V. V. , Yakovlev V. I. Basics of Classical Mechanics. - M.: Higher School, 1999. - S. 86-87. - ISBN 5-06-003587-5