# Dynamical, Hamiltonian and 2nd Order ODE Problems

Dynamical ordinary differential equations, such as those arising from the definition of a Hamiltonian system or a second order ODE, have a special structure that can be utilized in the solution of the differential equation. On this page we describe how to define second order differential equations for their efficient numerical solution.

## Mathematical Specification of a Dynamical ODE Problem

These algorithms require a Partitioned ODE of the form:

\[\frac{dv}{dt} = f_1(u,t) \\ \frac{du}{dt} = f_2(v) \\\]

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

and `f2(du,v,u,p,t)`

(in the inplace form), where `f1`

is independent of `v`

(unless specified by the solver), and `f2`

is independent of `u`

and `t`

. 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 `f2`

is a quadratic form. That means that `f2=v'*M*v`

, i.e. $\int f_2 = \frac{1}{2} m v^2$, 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(f::DynamicalODEFunction,v0,u0,tspan,p=NullParameters();kwargs...)
DynamicalODEProblem{isinplace}(f1,f2,v0,u0,tspan,p=NullParameters();kwargs...)
```

Defines the ODE with the specified functions. `isinplace`

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

Parameters are optional, and if not given then a `NullParameters()`

singleton will be used which will throw nice errors if you try to index non-existent parameters. Any extra keyword arguments are passed on to the solvers. For example, if you set a `callback`

in the problem, then that `callback`

will be added in every solve call.

### Fields

`f1`

and`f2`

: The functions in the ODE.`v0`

and`u0`

: The initial conditions.`tspan`

: The timespan for the problem.`p`

: The parameters for the problem. Defaults to`NullParameters`

`kwargs`

: The keyword arguments passed onto the solves.

## 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 $u_0$ which define an ODE:

\[u'' = f(u',u,p,t)\]

`f`

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

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

), 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:

\[v' = f(v,u,p,t) \\ u' = v \\\]

is generated.

### Constructors

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

Defines the ODE with the specified functions.

### Fields

`f`

: The function for the second derivative.`du0`

: The initial derivative.`u0`

: The initial condition.`tspan`

: The timespan for the problem.`callback`

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

## 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:

\[H(p,q)\]

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

### Constructors

`HamiltonianProblem{T}(H,p0,q0,tspan,param=nothing;kwargs...)`

### Fields

`H`

: The Hamiltonian`H(p,q,params)`

which returns a scalar.`p0`

: The initial momentums.`q0`

: The initial positions.`tspan`

: The timespan for the problem.`param`

: Defaults to`nothing`

.`param`

will be passed to`H`

's`params`

.