A Brief Introduction to Celestial Mechanics

Home , Conservative force

Harini Chandramouli

University of Minnesota

Climate Seminar October 2, 2018 Outline

1. History

2. The Kepler Problem

3. Conserved Quantities & Orbital Elements

4. Regularization

5. Other Important Topics

2 / 24 History Table of Contents

1. History

2. The Kepler Problem

3. Conserved Quantities & Orbital Elements

4. Regularization

5. Other Important Topics

3 / 24 A deﬁnition we care about more: 3 Celestial mechanics is the study of point particles in R (mostly) moving under the inﬂuence of their mutual gravitational attraction. Generally there is an emphasis on the general orbital motions of the solar system bodies.

History Deﬁnition

Celestial mechanics is considered to be the study of the motion of objects in outer space. Historically, celestial mechanics applies the principles of physics (classical mechanics) to astronomical objects (e.g. stars, planets) to acquire information on these objects.

4 / 24 3 Celestial mechanics is the study of point particles in R (mostly) moving under the inﬂuence of their mutual gravitational attraction. Generally there is an emphasis on the general orbital motions of the solar system bodies.

History Deﬁnition

A deﬁnition we care about more:

4 / 24 History Deﬁnition

A deﬁnition we care about more: 3 Celestial mechanics is the study of point particles in R (mostly) moving under the inﬂuence of their mutual gravitational attraction. Generally there is an emphasis on the general orbital motions of the solar system bodies.

4 / 24 History Timeline

Alexandria, ∼140CE, Ptolemy Poland, 1473 - 1543, Nicolaus Copernicus Denmark, 1546 - 1602, Tycho Brahe Germany, 1571 - 1630, Johannes Kepler England, 1643 - 1727, Isaac Newton

5 / 24 History Timeline

Switzerland, 1707 - 1783, Leonard Euler Sardinia, 1736 - 1813, Joseph-Louis Lagrange Prussia, 1804 - 1851, Carl Jacobi USA, 1838 - 1914, George Hill France, 1854 - 1912, Henri Poincare

6 / 24 The Kepler Problem Table of Contents

1. History

2. The Kepler Problem

3. Conserved Quantities & Orbital Elements

4. Regularization

5. Other Important Topics

7 / 24 The Kepler Problem The Two-Body Problem

Let us start by looking at a simple case - the two - body problem. The two-body problem is to determine the position and speed of two bodies interacting with each other given their masses, initial positions, and initial velocities. The gravitational two-body problem is a special case in which the two-bodies interact by a central force F , that varies in strength as the inverse of the distance, r, between them.

m1, x1 m2 r m2, x2 m1 Gm m F = 1 2 r2

8 / 24 The Kepler Problem The Two-Body Problem

The diﬀerential equations for the gravitational two-body problem are  x − x m x¨ = F · 2 1  1 1 r x − x m x¨ = F · 1 2  2 2 r with initial conditions

x1(t0), x2(t0), x˙1(t0), x˙2(t0)

for some initial time t0.

9 / 24 Then, we get the single second order ODE q q¨ = −G(m + m ) . 1 2 |q|3

This is called the Kepler problem. Now we turn our second order ODE into a system of ﬁrst order ODEs and we have the familiar  q˙ = p µq p˙ = −  |q|3

The Kepler Problem The Kepler Problem

We can reduce the two-body problem into a simpler problem by letting

q = x2 − x1.

10 / 24 Now we turn our second order ODE into a system of ﬁrst order ODEs and we have the familiar  q˙ = p µq p˙ = −  |q|3

The Kepler Problem The Kepler Problem

We can reduce the two-body problem into a simpler problem by letting

q = x2 − x1.

Then, we get the single second order ODE q q¨ = −G(m + m ) . 1 2 |q|3

This is called the Kepler problem.

10 / 24 The Kepler Problem The Kepler Problem

We can reduce the two-body problem into a simpler problem by letting

q = x2 − x1.

Then, we get the single second order ODE q q¨ = −G(m + m ) . 1 2 |q|3

This is called the Kepler problem. Now we turn our second order ODE into a system of ﬁrst order ODEs and we have the familiar  q˙ = p µq p˙ = −  |q|3

10 / 24 Conserved Quantities & Orbital Elements Table of Contents

1. History

2. The Kepler Problem

3. Conserved Quantities & Orbital Elements

4. Regularization

5. Other Important Topics

11 / 24 Conserved Quantities & Orbital Elements Conserved Quantities

A conserved quantity of a dynamical system is a function of the dependent variables whose value remains constant along each trajectory of the system.

Note: Emmy Noether’s ﬁrst theorem states that every diﬀerentiable symmetry of the action of a physical system has a corresponding conservation law. All the conserved quantities in the Kepler problem relate to a symmetry in the system.

12 / 24 We can then show that d (q × p) = q × p˙ + p × p = −µ|q|−3(q × q) + p × p = 0. dt So, we have q × p = c, a constant vector. We will refer to c as the angular momentum.

Conserved Quantities & Orbital Elements The Conservation of Angular Momentum

Angular momentum is the rotational analog of linear momentum. It is deﬁned for a point particle to be the vector

q × p.

13 / 24 So, we have q × p = c, a constant vector. We will refer to c as the angular momentum.

Conserved Quantities & Orbital Elements The Conservation of Angular Momentum

Angular momentum is the rotational analog of linear momentum. It is deﬁned for a point particle to be the vector

q × p.

We can then show that d (q × p) = q × p˙ + p × p = −µ|q|−3(q × q) + p × p = 0. dt

13 / 24 Conserved Quantities & Orbital Elements The Conservation of Angular Momentum

Angular momentum is the rotational analog of linear momentum. It is deﬁned for a point particle to be the vector

q × p.

We can then show that d (q × p) = q × p˙ + p × p = −µ|q|−3(q × q) + p × p = 0. dt So, we have q × p = c, a constant vector. We will refer to c as the angular momentum.

13 / 24 Integrate both sides and you’ll see that 1 µ |p|2 = + h, 2 |q| where h is our constant of integration and also our energy.

Note: Hamiltonians!

Conserved Quantities & Orbital Elements The Conservation of Energy

The gravitational interaction as described earlier was an internal conservative force and as such we can conclude that the energy is conserved in this system. We can can ﬁnd an expression for this mathematically as follows: µq p · p˙ = p · − |q|3 p · p˙ = −µ|q|−3(q · q˙) dq p · p˙ = −µ|q|−2 dt

14 / 24 Note: Hamiltonians!

Conserved Quantities & Orbital Elements The Conservation of Energy

14 / 24 Conserved Quantities & Orbital Elements The Conservation of Energy

Note: Hamiltonians! 14 / 24 Conserved Quantities & Orbital Elements Orbital Elements

eccentricity vector, eccentricity (e) semi-major axis (a) argument of the periapsis (ω) true anomaly (ν) inclination (i)

Figure: orbital plane, yellow, intersects reference plane, grey (wikipedia)

15 / 24 Conserved Quantities & Orbital Elements Another Conserved Quantity!

Another constant of motion is actually the eccentric axis, e. It can be show using vector identities that d q c × q = . dt |q| |q|3

Then multiplying both sides of this equation by −µ we can derive the following: d q µ = p˙ × c. dt |q| Integrating this statement, we get q µ e + = p × c, |q|

where e is the constant of integration.

16 / 24 The conic particle moves on a conic section of eccentricity e with one focus at the origin → this is Kepler’s ﬁrst law!

Conserved Quantities & Orbital Elements The Solution to Kepler’s Problem

This can be derived by taking the dot product of both sides of our last equation with q, then changing coordinates (polar coordinates), with coordinates (|q|, ν). We ﬁnd that the solution is given by

|c|2/µ |q| = . 1 + e cos(ν)

17 / 24 Conserved Quantities & Orbital Elements The Solution to Kepler’s Problem

|c|2/µ |q| = . 1 + e cos(ν)

The conic particle moves on a conic section of eccentricity e with one focus at the origin → this is Kepler’s ﬁrst law!

17 / 24 Regularization Table of Contents

1. History

2. The Kepler Problem

3. Conserved Quantities & Orbital Elements

4. Regularization

5. Other Important Topics

18 / 24 A singularity t0 of the Kepler problem is a collision singularity when q(t) approaches a speciﬁc point of ∆ as t → t0.

A regularization is a transformation (q, p) → (u, w) where q = 0 corresponds to some u = u0 and |w(s)| → w0 as s → s0.

Regularization Regularization

The collision set, i.e.

m2 ∆ = q |q| = 0 , |q| is the set of points where the m1 distance to the origin is 0.

19 / 24 A singularity t0 of the Kepler problem is a collision singularity when q(t) approaches a speciﬁc point of ∆ as t → t0.

A regularization is a transformation (q, p) → (u, w) where q = 0 corresponds to some u = u0 and |w(s)| → w0 as s → s0.

Regularization Regularization

The collision set, i.e.

m2 ∆ = q |q| = 0 , |q| is the set of points where the m1 distance to the origin is 0.

19 / 24 A regularization is a transformation (q, p) → (u, w) where q = 0 corresponds to some u = u0 and |w(s)| → w0 as s → s0.

Regularization Regularization

The collision set, i.e.

m2 ∆ = q |q| = 0 , |q| is the set of points where the m1 distance to the origin is 0.

A singularity t0 of the Kepler problem is a collision singularity when q(t) approaches a speciﬁc point of ∆ as t → t0.

19 / 24 Regularization Regularization

The collision set, i.e.

m2 ∆ = q |q| = 0 , |q| is the set of points where the m1 distance to the origin is 0.

A singularity t0 of the Kepler problem is a collision singularity when q(t) approaches a speciﬁc point of ∆ as t → t0.

A regularization is a transformation (q, p) → (u, w) where q = 0 corresponds to some u = u0 and |w(s)| → w0 as s → s0.

19 / 24 The Levi-Civita transformation is  q = 2z2 w p =  z¯ where q, p, z, w ∈ C. We can also utilize the use of a time change.

New diﬀerential equations:  w z0 = 2 w0 = hz where |p|2 µ |w|2 µ h = − = − . 2 |q| 2|z|2 2|z|2

Regularization Levi-Civita Regularization

2 ∼ Consider the problem in R = C, that is, instead of the vector 2 q = (q1, q2) ∈ R we use the complex notation q = q1 + iq2 ∈ C.

20 / 24 New diﬀerential equations:  w z0 = 2 w0 = hz where |p|2 µ |w|2 µ h = − = − . 2 |q| 2|z|2 2|z|2

Regularization Levi-Civita Regularization

2 ∼ Consider the problem in R = C, that is, instead of the vector 2 q = (q1, q2) ∈ R we use the complex notation q = q1 + iq2 ∈ C.

The Levi-Civita transformation is  q = 2z2 w p =  z¯ where q, p, z, w ∈ C. We can also utilize the use of a time change.

20 / 24 Regularization Levi-Civita Regularization

2 ∼ Consider the problem in R = C, that is, instead of the vector 2 q = (q1, q2) ∈ R we use the complex notation q = q1 + iq2 ∈ C.

The Levi-Civita transformation is  q = 2z2 w p =  z¯ where q, p, z, w ∈ C. We can also utilize the use of a time change.

New diﬀerential equations:  w z0 = 2 w0 = hz where |p|2 µ |w|2 µ h = − = − . 2 |q| 2|z|2 2|z|2

20 / 24 What is this regularization doing? Removes the singularity at the origin Singularities now lie at inﬁnity Double cover Go around once in the z-plane → go around twice in the q-plane Bounce

Regularization Levi-Civita Regularization

 w z0 = 2 w0 = hz

21 / 24 Regularization Levi-Civita Regularization

What is this regularization doing? Removes the singularity at the origin  w Singularities now lie at inﬁnity z0 = 2 Double cover w0 = hz Go around once in the z-plane → go around twice in the q-plane Bounce

21 / 24 Other Important Topics Table of Contents

1. History

2. The Kepler Problem

3. Conserved Quantities & Orbital Elements

4. Regularization

5. Other Important Topics

22 / 24 n-body problem Hamiltonian systems Lagrangian points Modern celestial mechanics

Other Important Topics Other Important Topics

23 / 24 Hamiltonian systems Lagrangian points Modern celestial mechanics

Other Important Topics Other Important Topics

n-body problem

23 / 24 Lagrangian points Modern celestial mechanics

Other Important Topics Other Important Topics

n-body problem Hamiltonian systems

23 / 24 Modern celestial mechanics

Other Important Topics Other Important Topics

n-body problem Hamiltonian systems Lagrangian points

23 / 24 Other Important Topics Other Important Topics

n-body problem Hamiltonian systems Lagrangian points Modern celestial mechanics

23 / 24 Other Important Topics

Thank You!

24 / 24