# Dynamical, Hamiltonian and 2nd Order ODE Problems

## Mathematical Specification of a Dynamical ODE Problem

These algorithms require a Partitioned ODE of the form:

This is a Partitioned ODE partitioned into two groups, so the functions should be specified as `f1(t,u,v,dx)`

and `f2(t,u,v,dv)`

(in the inplace form), where `f1`

is independent of `t`

and `u`

, and unless specified by the solver, `f2`

is independent of `v`

. This includes discretizations arising from `SecondOrderODEProblem`

s where the velocity is not used in the acceleration function, and Hamiltonians where the potential is (or can be) time-dependent but the kinetic energy is only dependent on `v`

.

Note that some methods assume that the integral of `f1`

is a quadratic form. That means that `f1=v'*M*v`

, i.e. $\int f_1 = \frac{1}{2} m v^2$, giving `du = v`

. This is equivalent to saying that the kinetic energy is related to $v^2$. The methods which require this assumption will lose accuracy if this assumption is violated. Methods listed make note of this requirement with "Requires quadratic kinetic energy".

### Constructor

`DynamicalODEProblem{isinplace}(f1,f2,u0,v0,tspan,callback=CallbackSet(),mass_matrix=I)`

Defines the ODE with the specified functions. `isinplace`

optionally sets whether the function is inplace or not. This is determined automatically, but not inferred.

### Fields

`f1`

and`f2`

: The functions in the ODE.`u0`

: The initial condition.`du0`

: The initial derivative.`tspan`

: The timespan for the problem.`callback`

: A callback to be applied to every solver which uses the problem. Defaults to nothing.`mass_matrix`

: The mass-matrix. Defaults to`I`

, the`UniformScaling`

identity matrix.

## Mathematical Specification of a 2nd Order ODE Problem

To define a 2nd Order ODE Problem, you simply need to give the function $f$ and the initial condition $u₀$ which define an ODE:

`f`

should be specified as `f(t,u,du)`

(or in-place as `f(t,u,du,ddu)`

), and `u₀`

should be an AbstractArray (or number) whose geometry matches the desired geometry of `u`

. Note that we are not limited to numbers or vectors for `u₀`

; one is allowed to provide `u₀`

as arbitrary matrices / higher dimension tensors as well.

From this form, a dynamical ODE:

is generated.

### Constructors

`SecondOrderODEProblem{isinplace}(f,u0,du0,tspan,callback=CallbackSet(),mass_matrix=I)`

Defines the ODE with the specified functions.

### Fields

`f`

: The function in the ODE.`u0`

: The initial condition.`du0`

: The initial derivative.`tspan`

: The timespan for the problem.`callback`

: A callback to be applied to every solver which uses the problem. Defaults to nothing.`mass_matrix`

: The mass-matrix. Defaults to`I`

, the`UniformScaling`

identity matrix.

## Hamiltonian Problems

`HamiltonianProblem`

s are provided by DiffEqPhysics.jl and provide an easy way to define equations of motion from the corresponding Hamiltonian. To define a `HamiltonianProblem`

one only needs to specify the Hamiltonian:

and autodifferentiation (via ForwardDiff.jl) will create the appropriate equations.

### Constructors

`HamiltonianProblem{T}(H,q0,p0,tspan;kwargs...)`

### Fields

`H`

: The Hamiltonian`H(p,q)`

which returns a scalar.`q0`

: The initial positions.`p0`

: The initial momentums.`tspan`

: The timespan for the problem.`callback`

: A callback to be applied to every solver which uses the problem. Defaults to nothing.`mass_matrix`

: The mass-matrix. Defaults to`I`

, the`UniformScaling`

identity matrix.