To solve this problem, let's look at the system step-by-step.
### Step 1: Define the Parameters
We need to consider the following parameters:
- [tex]\( G \)[/tex] is the gravitational constant.
- [tex]\( m \)[/tex] is the mass of the planet.
- [tex]\( M \)[/tex] is the mass of the star.
- [tex]\( r \)[/tex] is the orbital radius.
- [tex]\( L \)[/tex] is the angular momentum of the planet.
### Step 2: Kinetic Energy (T)
For spherical coordinates, and considering only radial and angular components (ignoring the motion along the z-axis), the kinetic energy of the planet in its orbit around the star can be expressed as:
[tex]\[ T = \frac{L^2}{2 m r^2} \][/tex]
### Step 3: Potential Energy (U)
The potential energy between the planet and the star, as given by the problem, is:
[tex]\[ U = -\frac{G m M}{r} \][/tex]
### Step 4: Lagrangian (L)
The Lagrangian [tex]\( \mathcal{L} \)[/tex] of a system is defined as the difference between the kinetic and potential energies:
[tex]\[ \mathcal{L} = T - U \][/tex]
Substitute the expressions for [tex]\( T \)[/tex] and [tex]\( U \)[/tex]:
[tex]\[ \mathcal{L} = \frac{L^2}{2 m r^2} - \left(-\frac{G m M}{r}\right) \][/tex]
[tex]\[ \mathcal{L} = \frac{L^2}{2 m r^2} + \frac{G m M}{r} \][/tex]
### Step 5: Hamiltonian (H)
The Hamiltonian [tex]\( H \)[/tex] is a conserved quantity in the absence of explicit time dependence and is equal to the total energy of the system, which is the sum of the kinetic and potential energies:
[tex]\[ H = T + U \][/tex]
Substitute the expressions for [tex]\( T \)[/tex] and [tex]\( U \)[/tex]:
[tex]\[ H = \frac{L^2}{2 m r^2} + \left(-\frac{G m M}{r}\right) \][/tex]
[tex]\[ H = \frac{L^2}{2 m r^2} - \frac{G m M}{r} \][/tex]
### Step 6: Angular Momentum
Angular momentum [tex]\( L \)[/tex] of the planet is also a conserved quantity in this system.
### Final Answer:
After following the above steps, we have:
- Lagrangian:
[tex]\[ \mathcal{L} = \frac{L^2}{2 m r^2} + \frac{G m M}{r} \][/tex]
- Hamiltonian:
[tex]\[ H = \frac{L^2}{2 m r^2} - \frac{G m M}{r} \][/tex]
- Conserved Quantities:
- Hamiltonian (Total Energy)
- Angular Momentum
### Conclusion:
The conserved quantities of this system are the Hamiltonian (which represents the total energy) and the angular momentum. The Lagrangian and Hamiltonian for the system are as derived above.
The resulting expressions for these quantities are:
[tex]\[ \text{Lagrangian} = \frac{L^2}{2 m r^2} + \frac{G m M}{r} \][/tex]
[tex]\[ \text{Hamiltonian} = \frac{L^2}{2 m r^2} - \frac{G m M}{r} \][/tex]
[tex]\[ \text{Angular Momentum} = L \][/tex]
