# A particle of mass M moves in a plane under the influence of a force F = – kx directed towards the origin. What is the Lagrangian of the system in polar coordinate (r, θ)

1. $\mathbf{M\left&space;(&space;\frac{dr}{dt}&space;\right&space;)^{2}+&space;\frac{1}{2}kr^{2}}$
2. $\mathbf{M\left&space;(&space;\frac{dr}{dt}&space;\right&space;)^{2}&space;-&space;\frac{1}{2}kr^{2}}$
3. $\mathbf{\frac{1}{2}M\left&space;[&space;\left&space;(&space;\frac{dr}{dt}&space;\right&space;)^{2}&space;+&space;r^{2}\left&space;(&space;\frac{d\theta&space;}{dt}&space;\right&space;)^{2}&space;\right&space;]&space;+&space;\frac{1}{2}kr^{2}}$
4. $\mathbf{\frac{1}{2}M\left&space;[&space;\left&space;(&space;\frac{dr}{dt}&space;\right&space;)^{2}&space;+&space;r^{2}\left&space;(&space;\frac{d\theta&space;}{dt}&space;\right&space;)^{2}&space;\right&space;]&space;-&space;\frac{1}{2}kr^{2}}$

## A particle of mass M moves in a plane under the influence of a force F = – kx directed towards the origin. What is the Lagrangian of the system in polar coordinate (r, θ)

A particle of mass M moves in a plane under the influence of a force F = – kx directed towards the origin. What is the Lagrangian of the system in polar coordinate (r, θ):

$\mathbf{\frac{1}{2}M\left&space;[&space;\left&space;(&space;\frac{dr}{dt}&space;\right&space;)^{2}&space;+&space;r^{2}\left&space;(&space;\frac{d\theta&space;}{dt}&space;\right&space;)^{2}&space;\right&space;]&space;-&space;\frac{1}{2}kr^{2}}$

or

$\mathbf{L&space;=&space;\frac{1}{2}M\left&space;[&space;(\dot{r})^{2}&space;+&space;r^{2}(\dot{\theta&space;})^{2}&space;\right&space;]&space;-&space;\frac{1}{2}kr^{2}}$

When a particle is moving under a central force, then the force is conservative and the motion is in a plane.

Kinetic energy

$\mathbf{T&space;=&space;\frac{1}{2}M\left&space;[&space;\left&space;(&space;\frac{dr}{dt}&space;\right&space;)^{2}&space;+&space;r^{2}\left&space;(&space;\frac{d\theta&space;}{dt}&space;\right&space;)^{2}&space;\right&space;]}$

or

$\mathbf{T&space;=&space;\frac{1}{2}M\left&space;[&space;\left&space;(&space;\dot{r}&space;\right&space;)^{2}&space;+&space;r^{2}&space;(&space;\dot{\theta&space;}&space;)^{2}&space;\right&space;]}$

Potential energy

$\mathbf{F&space;=&space;-&space;kr}$

Relation between force and potential energy.

$\mathbf{F&space;=&space;-\frac{dV}{dr}}$

$\mathbf{V&space;=&space;-&space;\int&space;F&space;dr}$

$\mathbf{V&space;=&space;-&space;\int&space;(-kr)&space;dr}$

$\mathbf{V&space;=&space;k&space;\int&space;r&space;dr}$

$\mathbf{V&space;=&space;\frac{1}{2}kr^{2}}$

Lagrangian of the system

$\mathbf{L&space;=&space;T&space;-&space;V}$

$\mathbf{L&space;=&space;\frac{1}{2}M\left&space;[&space;\left&space;(&space;\frac{dr}{dt}&space;\right&space;)^{2}&space;+&space;r^{2}\left&space;(&space;\frac{d\theta&space;}{dt}&space;\right&space;)^{2}&space;\right&space;]&space;-&space;\frac{1}{2}kr^{2}}$

or

$\mathbf{L&space;=&space;\frac{1}{2}M\left&space;[&space;(\dot{r})^{2}&space;+&space;r^{2}(\dot{\theta&space;})^{2}&space;\right&space;]&space;-&space;\frac{1}{2}kr^{2}}$

 MSC Physics MCQs 1. Atomic Physics 2. Nuclear Physics 3. Classical Mechanics 4. Quantum Mechanics 5. Electronics Physics 6. Solid State Physics 7. Physical Wave Optics 8. Waves and Oscillations 9. Electromagnetic Theory 10. Special Theory of Relativity 11. Heat and Thermodynamics 12. Mathematical Method for Physics