It is the most succinct introduction to Noether's theorem I've ever seen online. Your lectures are almost comparable to Lenoard Susskind's Classical mechanics series, except you go deeper with more elaborations on the derivations of the formulations. Many thanks for the excellent work !
13:11 I've got this by calculating the z component of the angular momentum of the particle about the origin using standard basis vectors i.e. z-component of (r X dr/dt) = z-component of (xi+yj+zk) X (dx/dt i + dy/dt j + dz/dt k) = x dy/dt- y dx/dt and then changing to spherical coords. Also, this is the same as the z component of the angular momentum of the particle about the z-axis i.e. z-component of (xi+yj) X (dx/dt i + dy/dt j + dz/dt k) = x dy/dt- y dx/dt. Intuitively, if θ and r are constant, then the velocity of the particle using spherical basis vectors e_φ, e_r and e_θ is (rsinθ dφ/dt)e_φ and the result follows.
It is the most succinct introduction to Noether's theorem I've ever seen online. Your lectures are almost comparable to Lenoard Susskind's Classical mechanics series, except you go deeper with more elaborations on the derivations of the formulations. Many thanks for the excellent work !
I honestly wish I can give you more than one like, this was really helpful. Thank you very much for this well explained lesson.
Thank you so much sir
13:11 I've got this by calculating the z component of the angular momentum of the particle about the origin using standard basis vectors i.e.
z-component of (r X dr/dt) = z-component of (xi+yj+zk) X (dx/dt i + dy/dt j + dz/dt k) = x dy/dt- y dx/dt and then changing to spherical coords.
Also, this is the same as the z component of the angular momentum of the particle about the z-axis i.e.
z-component of (xi+yj) X (dx/dt i + dy/dt j + dz/dt k) = x dy/dt- y dx/dt.
Intuitively, if θ and r are constant, then the velocity of the particle using spherical basis vectors e_φ, e_r and e_θ is (rsinθ dφ/dt)e_φ
and the result follows.
How distance betwen moon and planet can depend of placemant of origin of coord. System?
It doesn't. Choice of origin is a gauge choice.