##### Level 1: One body, one dimensional motion and static potential

Description: Below are trajectories x(t) of a point mass in a unspecified potential well V(x), use any means to determine the potential.

Hint: the equation of motion is described by the Hamiltonian H=p^2/2 + V(x)

Case 1: trajectory and data Ans: V(x)=x^2/2 (**Solved by Lambert Kong, 1/12/2024**)

Case 2: trajectory and data Ans: V(x)=-x^2/2+x^4 (**Solved by Lambert Kong, 1/12/2024**)

Case 3: trajectory and data Ans: V(x)=x^6/6 (**Solved by Lambert Kong, 1/12/2024**)

##### Level 2: Two bodies, two dimensional motion and static interactions

Description: Below are trajectories (x1,y1) and (x2,y2) of two particles with masses 5 and 1. They interact with each other in free space with a potential V(r) that depends on their separation r=[(x1-x2)^2+(y1-y2)^2]^0.5. Determine V(r).

Hint: the motion is described by the Hamiltonian H=p1^2/5+p2^2+V(r), where pi is the 2D momentum of the i-th particle.

Case 1: trajectory and data4.cvs Ans: V(r)=-1/r^2 (**Solved by Ahan Datta, 1/14/2024**)

Case 2: trajectory and data5.cvs Ans: V(r)=r^6 (**Solved by Ahan Datta, 1/16/2024**)

##### Special challenge:

Description: Christopher Moore discovered the first stable, periodic three-body motion, see Physical Review Letters, 70 3675 (1993). It looks like an infinity symbol. The Hamiltonian is given by H=p1^2+p2^2+p3^2+V(r12)+V(r23)+V(r13), where r12 is the distance between the 1st and the 2nd particle and so on. Determine the interaction potential V(r).

Trajectory with perfect initial condition and data6.cvs Ans: V(r)=-1/2r (**Solved by Ahan Datta, 1/18/2024**)

Trajectory with slightly off initial condition and data7.cvs

##### Level 3: EoS of two bodies in two dimensional motion with conservative potential

Description: Below are trajectories (x1,y1) and (x2,y2) of two particles with equal masses. They interact with each other in free space with a conservative potential V. Determine the equation of state EoS expressed in terms of their positions (x_i, y_i), and their time derivatives.

Hint 1: If we impose the knowledge of the Hamiltonian H=p1^2+p2^2+V(r), we can solve it the old way. Here, without the knowledge of H, we are seeking the underlying differential equations expressed in terms of their position, velocities and accelerations.

Simplification1: The interaction depends on their separation r=[(x1-x2)^2+(y1-y2)^2]^0.5 as V(r)=A r^alpha + B r^beta, where A, B, alpha and beta are constants to be determined.

Simplification 2: You can assume no velocity terms in the differential equations.

Case 1: trajectory and data8.cvs (based on standard molecular potential model)

Case 2: trajectory and data9.cvs

Practice case: Determine the 2 differential equations the govern the trajectory (x[t], y[t]). The equations follow the general form

x'[t]+A x[t]^alpha + B y[t]^beta=0

y'[t]+C x[t]^gamma + D y[t]^delta=0,

where A, B, C, D, alpha, beta, gamma and delta are all constants.

trajectory and data10.cvs Ans: dx/dt-x^2+2y=0, dy/dt-2x+y^2=0 (**Solved by Jarvis Zhang, 1/23/2024**)

##### Level 4: Learning and applying physics concept

Description: N particles are interacting in free space with pair-wise interaction potential V_ij= V(x_i, x_j), where x_i is the coordinate of the i-th particle. The Hamiltonian can be written as H=p1^2+p^2/m2+...pN^2/mN+V_12+V_13+...+V_{N-1,N}. Note that we assume m1=1. Our goal is to learn about the potential with simple examples and then determine the masses m2, m3.... in a complex system. (This is how Neptune and Pluto were discovered.)

**Step 0**: We may start with the brutal force approach directly on N=4 particles and see if your program can identify the underlying equation of motion for all the particles and determine m2, m3 and m4.

trajectory and data17.cvs

**Step 1**: Alternatively we start with the training stage based on the following 2-body exercises. Solve their equations and identify the mass m2 for the following 3 cases.

trajectoryA and data12.cvs Ans: V(r)=-1/r and masses are 2:1 (**Solved by Ahan Datta, 2/16/2024**)

**Step 2**: Let your program "learn" from the above examples, and attempt to solve the following 3-body problems

Determine m2 and m3 in the 3-body system:

trajectory and data15.cvs

trajectory and data16.cvs

**Step 3**: Check if your program can solve the 4-body problem in step 0.

Submit your answer to the AI-Physics Slack channel or email Lambert <lambertkk@uchicago.edu>