Symplectic Ball Packing

Fix $n=2$ and take $\epsilon \in (0,1)$. For $i = 1,\dots,4$, let $B_i \subset \R^{2n}$ denote the closed unit ball centered at $(3i-3,0,\dots,0)$. Your goal is to find an algorithm that takes $\epislon$ as input and produces as output an explicit smooth Hamiltonian function $H_\epsilon: \R^{2n} \times [0,1] \rightarrow \R$ whose time-$1$ Hamiltonian flow $\phi: \R^{2n} \ra \R^{2n}$ satisfies $\phi(\sqcup_{i=1}^{4}B_i) \subset B^{2n}(R)$, where $B^{2n}(R) \subset \R^{2n}$ is the closed ball centered at the origin of radius $R$, and where we require the volume ratio of $\bigsqcup\limits_{i=1}^{4} B_i$ to $B^{2n}(R)$ to satisfy $\dfrac{4}{R^{2n}} > 1 - \epsilon$. Present the algorithm as a Python program that takes $\epislon$ as input. The algorithm's output must itself be a Python program that implements the Hamiltonian for the given $\epislon$. This program should be in a file called Hamiltonian.py which implements a function Hamiltonian(Q, P, t) that has the following properties. - Takes inputs Q (positions), P (momenta), and t (time). - Must be implemented using PyTorch operations to support automatic differentiation. - Inputs Q and P will be `torch.Tensor`s of shape `(N, n)` (where N is the size of a batch size of points sampled from the boundaries of the input balls). - Input t will be a scalar float. - Returns a `torch.Tensor` of shape `(N,)` representing the Hamiltonian value for each point in the batch. - Is smooth, to ensure existence and uniqueness of flow. A solution to this problem is meant to rest on finding a family of embeddings that has a natural algorithmic dependence on $\epsilon$. As such, the algorithm that produces the Hamiltonians for a given $\epsilon$ must finish running in under an hour on a typical laptop. The Hamiltonian function itself will be called multiple times during solution verification and must return within ten seconds on each call.

Fix $n=2$. Let $k \ge 10$ be an integer and take $\epsilon \in (0,1)$. For $i = 1,\dots,k$, let $B_i \subset \R^{2n}$ denote the closed unit ball centered at $(3i-3,0,\dots,0)$. Your goal is to find an algorithm that takes $k$ and $\epislon$ as inputs and produces as output an explicit smooth Hamiltonian function $H_\epsilon: \R^{2n} \times [0,1] \rightarrow \R$ whose time-$1$ Hamiltonian flow $\phi: \R^{2n} \ra \R^{2n}$ satisfies $\phi(\sqcup_{i=1}^{k}B_i) \subset B^{2n}(R)$, where $B^{2n}(R) \subset \R^{2n}$ is the closed ball centered at the origin of radius $R$, and where we require the volume ratio of $\bigsqcup\limits_{i=1}^{k} B_i$ to $B^{2n}(R)$ to satisfy $\dfrac{k}{R^{2n}} > 1 - \epsilon$. Present the algorithm as a Python program that takes $k$ and $\epislon$ as inputs. The algorithm's output must itself be a Python program that implements the Hamiltonian for the given $k$ and $\epislon$. This program should be in a file called Hamiltonian.py which implements a function Hamiltonian(Q, P, t) that has the following properties. - Takes inputs Q (positions), P (momenta), and t (time). - Must be implemented using PyTorch operations to support automatic differentiation. - Inputs Q and P will be `torch.Tensor`s of shape `(N, n)` (where N is the size of a batch size of points sampled from the boundaries of the input balls). - Input t will be a scalar float. - Returns a `torch.Tensor` of shape `(N,)` representing the Hamiltonian value for each point in the batch. - Is smooth, to ensure existence and uniqueness of flow. A solution to this problem is meant to rest on finding a family of embeddings that has a natural algorithmic dependence on $\epsilon$. As such, the algorithm that produces the Hamiltonians for a given $\epsilon$ must finish running in under an hour on a typical laptop. The Hamiltonian function itself will be called multiple times during solution verification and must return within ten seconds on each call.