Proof -- motion under a central force in text Symon Mechanics

[kylinsky]

1. The derivation
In a 3-dim space,a particle is acted by a central force(the center of the force fixed in the origin) .we now take the motion entirely in the xy-plane and write the equations of the motion in polar coordinate

how can i derive from these equation that
T(kinetic energy)+V(potential)=E=

?
(sorry for my poor english)

Homework Equations

The Attempt at a Solution

 

[ehild]

1. The derivation
In a 3-dim space,a particle is acted by a central force(the center of the force fixed in the origin) .we now take the motion entirely in the xy-plane and write the equations of the motion in polar coordinate
View attachment 104870
how can i derive from these equation that
T(kinetic energy)+V(potential)=E=View attachment 104871?
(sorry for my poor english)

Read http://leandros.physics.uoi.gr/cm1/book-cm/ch6.pdf, for example.
You need to integrate both equations. The integral of the second one is conservation of angular momentum, that of the first one is conservation of energy.

 

[kylinsky]

Read http://leandros.physics.uoi.gr/cm1/book-cm/ch6.pdf, for example.
You need to integrate both equations. The integral of the second one is conservation of angular momentum, that of the first one is conservation of energy.

thanks!