Question

Obtain Hamilton's equations for a particle moving in space under an inverse- square-law central force, taking the \(q_{a}\) s to be spherical polar coordinates.

Short answer

Question: Derive Hamilton's equations for a particle moving under an inverse-square-law central force in spherical polar coordinates. Solution: 1. Write down the inverse-square-law force equation: \(F(r) = \dfrac{k}{r^2}\). 2. Find the particle's Lagrangian in spherical polar coordinates: \(L = \dfrac{1}{2} m \left(\dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2\theta \dot{\phi}^2\right) + \dfrac{k}{r}\). 3. Calculate the conjugate momenta: \[p_r = m\dot{r}\] \[p_\theta = m r^2 \dot{\theta}\] \[p_\phi = m r^2 \sin^2\theta \dot{\phi}\] 4. Derive the Hamiltonian: \(H = \dfrac{p_r^2}{2m} + \dfrac{p_\theta^2}{2mr^2} + \dfrac{p_\phi^2}{2mr^2 \sin^2\theta} - \dfrac{k}{r}\). 5. Obtain Hamilton's equations: \[\dfrac{\partial H}{\partial p_r} = \dot{r}\] \[-\dfrac{\partial H}{\partial r} = \dot{p_r}\] \[\dfrac{\partial H}{\partial p_\theta} = \dot{\theta}\] \[-\dfrac{\partial H}{\partial\theta} = \dot{p_\theta}\] \[\dfrac{\partial H}{\partial p_\phi} = \dot{\phi}\] \[-\dfrac{\partial H}{\partial\phi} = \dot{p_\phi}\]

Step-by-step solution

Inverse-square-law force equation

The inverse-square-law force equation is given by: \[F(r) = \dfrac{k}{r^2}\] where \(k\) is the force constant and \(r\) is the distance from the origin.

Lagrangian in spherical polar coordinates

To find the Lagrangian, first, we need to write down the kinetic and potential energies in spherical polar coordinates. The kinetic energy \(T\) is given by: \[T = \dfrac{1}{2} m \left(\dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2\theta \dot{\phi}^2\right)\] And the potential energy \(V\) is given by: \[V = -\dfrac{k}{r}\] Now, we can write the Lagrangian \(L\), which is the difference between the kinetic and potential energies: \[L = T - V= \dfrac{1}{2} m \left(\dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2\theta \dot{\phi}^2\right) + \dfrac{k}{r}\]

Calculate the conjugate momenta

To find the conjugate momenta, we must differentiate the Lagrangian with respect to the generalized velocities: \[\begin{aligned} p_r &= \dfrac{\partial L}{\partial \dot{r}} = m\dot{r} \\ p_\theta &= \dfrac{\partial L}{\partial \dot{\theta}} = m r^2 \dot{\theta} \\ p_\phi &= \dfrac{\partial L}{\partial \dot{\phi}} = m r^2 \sin^2\theta \dot{\phi} \end{aligned}\]

Derive the Hamiltonian

Now we derive the Hamiltonian \(H\) by performing the Legendre transformation, which is given by: \[H(q_a, p_a) = \sum_{a=1}^{3} p_a \dot{q_a} - L\] After substituting the expressions for the conjugate momenta and the Lagrangian, we get: \[H = \dfrac{p_r^2}{2m} + \dfrac{p_\theta^2}{2mr^2} + \dfrac{p_\phi^2}{2mr^2 \sin^2\theta} - \dfrac{k}{r}\]

Obtain Hamilton's equations

Finally, we can obtain Hamilton's equations for the system. There are two types of Hamilton's equations, which are given as follows: \[\begin{aligned} \dfrac{\partial H}{\partial p_a} &= \dot{q_a} \\ -\dfrac{\partial H}{\partial q_a} &= \dot{p_a} \end{aligned}\] Hence, we can differentiate the Hamiltonian with respect to the conjugate momenta and coordinates: \[\begin{aligned} \dfrac{\partial H}{\partial p_r} &= \dot{r} \\ -\dfrac{\partial H}{\partial r} &= \dot{p_r} \\ \dfrac{\partial H}{\partial p_\theta} &= \dot{\theta} \\ -\dfrac{\partial H}{\partial\theta} &= \dot{p_\theta} \\ \dfrac{\partial H}{\partial p_\phi} &= \dot{\phi}\\ -\dfrac{\partial H}{\partial\phi} &= \dot{p_\phi} \end{aligned}\] These are Hamilton's equations for a particle moving in space under an inverse-square-law central force in spherical polar coordinates.