Momentum (\(\mathbf{p}\)) is defined as the product of an object’s mass (\(m\)) and its velocity (\(\mathbf{v}\)):

\[ \mathbf{p} = m \mathbf{v} \]

The conservation of momentum follows directly from Newton’s Second and Third Laws:

1. Newton’s Second Law states that the rate of change of momentum is equal to the net force acting on a system:
   \[ \frac{d\mathbf{p}}{dt} = \mathbf{F} \]
   If no external force (\(\mathbf{F} = 0\)) acts on a system, then \(\frac{d\mathbf{p}}{dt} = 0\), meaning momentum is constant.
   
2. Newton’s Third Law (action-reaction principle) ensures that in an isolated system, internal forces between objects cancel out, so the total momentum remains unchanged.
   

• Collisions: Inelastic and elastic collisions both obey momentum conservation.
  
• Rocket propulsion: A rocket ejects mass (fuel exhaust) backward, conserving total momentum.
  
• Orbital mechanics: Planets and satellites follow paths governed by conservation of momentum and angular momentum.
  

In relativistic physics, momentum is also conserved, but it is modified by the factor from Einstein’s theory of relativity, using the relativistic momentum:

\[ \mathbf{p} = \gamma m \mathbf{v}, \quad \text{where } \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]

Thus, momentum conservation is a fundamental law that applies across classical, relativistic, and quantum mechanics.

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Can you go into more detail for orbital mechanics?

Orbital mechanics is the study of how objects move under the influence of gravity, particularly planets, moons, and satellites. The conservation of linear momentum and angular momentum plays a crucial role in describing these motions.

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The linear momentum of an orbiting body is given by:

\[ \mathbf{p} = m \mathbf{v} \]

Since gravity is a central force (it always points toward the center of mass of the system), it does not exert a force in the direction of motion (tangential direction). Instead, it changes the direction of the velocity, causing the body to move in a curved path (orbit), but it does not change the total linear momentum of the entire system.

For a two-body system, such as a planet orbiting a star, the center of mass (barycenter) follows a straight-line motion if no external force acts on it. The total linear momentum of the system remains conserved:

\[ m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2 = \text{constant} \]

In cases like the Earth-Sun system, the Sun moves slightly due to Earth’s gravitational pull, keeping the total momentum unchanged.

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Since gravity acts radially, it exerts no torque about the center of mass. This means that angular momentum is conserved:

\[ \mathbf{L} = \mathbf{r} \times \mathbf{p} = m \mathbf{r} \times \mathbf{v} \]

\[ \frac{d\mathbf{L}}{dt} = \mathbf{r} \times \mathbf{F} = 0 \]

which implies:

\[ \mathbf{L} = \text{constant} \]

One direct result of angular momentum conservation is Kepler’s Second Law, which states:

> A planet sweeps out equal areas in equal time intervals.

This means that when a planet is closer to the Sun (perihelion), it moves faster, and when it is farther (aphelion), it moves slower. This is due to the requirement that:

\[ m v r = \text{constant} \]

Since \( r \) is smaller at perihelion, \( v \) must increase to compensate, and vice versa.

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For Jupiter’s moons, each moon conserves its own angular momentum around Jupiter. If you select one moon’s radius and period as the unit system (as you mentioned wanting to do), you can compare the other moons’ motions using Kepler’s Third Law:

\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]

This law follows from the conservation of angular momentum and Newton’s Law of Universal Gravitation.

Would you like to explore an example of computing the motion of Jupiter’s moons using Julia?