# Class LagrangeRollerSim

Rollercoaster simulation that uses Lagrangian method of finding equations of motion. The shape of the roller coaster path is defined by HumpPath.

The Lagrangian method of finding the equations of motion is very different from the methods used in the other roller coaster simulations such as RollerSingleSim. For example, the NumericalPath is used here only for finding the initial conditions such as the path length position corresponding to the starting X value. Whereas in Roller1Sim the NumericalPath is used in the `evaluate()` method to find the rates of change.

## Variables and Parameters

The variables stored in the VarsList are:

``````vars[0] = x position of the ball
vars[1] = v = dx/dt = x velocity of the ball
vars[2] = s = position measured along length of track
vars[3] = s' = ds/dt = velocity measured along length of track
``````

The independent variables are the X position and X velocity. The position along the track `s` and velocity along the track `s'` are derived from the X position and X velocity of the ball.

## Parameters are: `text g = gravity m = mass ` Equation of Motion

The equation of the HumpPath is

``````y = 3 - (7/6) x^2 + (1/6) x^4
``````

The equations of motion are derived from the HumpPath as shown in the file [RollerCoaster Lagrangian(../RollerCoaster_Lagrangian.pdf). They turn out to be:

``````x' = v

-(x * (-7 + 2*x^2) * (3*g + (-7 + 6*x^2) * v^2))
v' = ------------------------------------------------
(9 + 49*x^2 - 28*x^4 + 4*x^6)
``````

## Track Position and Velocity

The track position `s` and velocity `s' = ds/dt` are derived from the X position and X velocity of the ball.

``````s = position along length of track
ds/dt = (ds/dx) (dx/dt)
s = integral (ds/dt) dt
s = integral (ds/dx) (dx/dt) dt
``````

A fundamental result of calculus relates the path length and slope of the path (this is only valid for curves where the slope is finite everywhere):

``````ds/dx = sqrt(1 + (dy/dx)^2)
``````

From the definition of the HumpPath we can easily find the slope `dy/dx` as a function of `x`:

``````dy/dx = -(7/3) x + (2/3) x^3
``````

Putting these together we get the track velocity as a function of `x` and `v`:

``````ds/dt = sqrt(1 + (-(7/3) x + (2/3) x^3)^2) v
``````

The track position `s` is then found by integrating `ds/dt` over time.

## Y Velocity

The vertical velocity of the ball, `y' = dy/dt`, can be found as a function of the independent variables `x` and `v` as follows:

``````dy/dt = (dy/dx) (dx/dt)
dy/dt = (-(7/3) x + (2/3) x^3) v
``````

Note that this should agree with:

``````s' = ds/dt = (+/-)sqrt((dx/dt)^2 + (dy/dt)^2)
``````

## Properties

initialState_: null | number[] = null

Initial values.

lowestPoint_: number

lowest possible y coordinate of path

potentialOffset_: number = 0

potential energy offset

## Methods

• Adds the given Observer to this Subject's list of Observers, so that the Observer will be notified of changes in this Subject. An Observer may call `Subject.addObserver` during its `observe` method.

#### Parameters

• ##### observer: Observer

the Observer to add

#### Returns void

• Adds the Parameter to the list of this Subject's available Parameters.

#### Parameters

• ##### parameter: Parameter

the Parameter to add

#### Throws

if a Parameter with the same name already exists.

• Notifies all Observers that the Parameter with the given name has changed by calling observe on each Observer.

#### Parameters

• ##### name: string

the language-independent or English name of the Parameter that has changed

#### Throws

if there is no Parameter with the given name

• Defines the differential equations of this ODESim; for an input set of variables, returns the current rate of change for each variable (the first derivative of each variable with respect to time).

The `timeStep` is the time since the state variables were last fully calculated, which can be and often is zero. The current time can be regarded as `getTime() + timeStep`. The input variables correspond to the Simulation state at that time. Note that `timeStep` is different from the time step used to advance the Simulation (as in AdvanceStrategy.advance). The `timeStep` is typically used when finding collisions in CollisionSim.findCollisions.

#### Parameters

• ##### vars: number[]

the current array of state variables (input), corresponding to the state at `getTime() + timeStep`

• ##### change: number[]

array of change rates for each variable (output), all values are zero on entry.

• ##### _timeStep: number

the current time step (might be zero)

#### Returns null | object

`null` if the evaluation succeeds, otherwise an object relating to the error that occurred. The `change` array contains the output results.

• Called at the end of a mouse drag operation, performs whatever action is appropriate. Only called if startDrag returned `true`.

#### Parameters

• ##### _simObject: null | SimObject

the SimObject being dragged, or `null` if no SimObject was found

• ##### _location: Vector

the location of the mouse in simulation coordinates of the SimView where `simObject` was found, or in the focus view if `simObject` is `null`.

• ##### _offset: Vector

distance from the initial object position to the mouse location at start of drag.

#### Returns void

• Returns the Parameter with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the Parameter

#### Returns Parameter

the Parameter with the given name

#### Throws

if there is no Parameter with the given name

• Returns the ParameterBoolean with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the ParameterBoolean

#### Returns ParameterBoolean

the ParameterBoolean with the given name

#### Throws

if there is no ParameterBoolean with the given name

• Returns the ParameterNumber with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the ParameterNumber

#### Returns ParameterNumber

the ParameterNumber with the given name

#### Throws

if there is no ParameterNumber with the given name

• Returns the ParameterString with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the ParameterString

#### Returns ParameterString

the ParameterString with the given name

#### Throws

if there is no ParameterString with the given name

• Returns the current Simulation time.

#### Returns number

the current Simulation time.

#### Throws

if there is no time variable for the simulation

• Called when a key is pressed or released, performs whatever action is appropriate for that event.

#### Parameters

• ##### _evt: KeyboardEvent

the KeyboardEvent that happened

• ##### _pressed: boolean

`true` means this is a key-down event; `false` means a key-up event

• ##### _modifiers: ModifierKeys

the modifier keys down during event

#### Returns void

• Called at each movement during a mouse drag, performs whatever action is appropriate. Only called if startDrag returned `true`. The SimObject being moved is passed in, along with the current mouse position, in simulation coordinates, and an offset calculated at the start of the drag.

Setting the SimObject position to `(x - offsetX, y - offsetY)` will move the SimObject smoothly along with the mouse movement.

#### Parameters

• ##### simObject: null | SimObject

the SimObject being dragged, or `null` if no SimObject was found

• ##### location: Vector

the location of the mouse in simulation coordinates of the SimView where `simObject` was found, or in the focus view if `simObject` is `null`.

• ##### offset: Vector

distance from the initial object position (from DisplayObject.getPosition) to the mouse location at start of drag.

#### Returns void

• Removes the Observer from this Subject's list of Observers. An Observer may call `removeObserver` during its `observe` method.

#### Parameters

• ##### observer: Observer

the Observer to detach from list of Observers

#### Returns void

• Removes the Parameter from the list of this Subject's available Parameters.

#### Parameters

• ##### parameter: Parameter

the Parameter to remove

#### Returns void

• Sets whether this Subject will broadcast events, typically used to temporarily disable broadcasting. Intended to be used in situations where a subclass overrides a method that broadcasts an event. This allows the subclass to prevent the superclass broadcasting that event, so that the subclass can broadcast the event when the method is completed.

#### Parameters

• ##### value: boolean

whether this Subject should broadcast events

#### Returns boolean

the previous value

• Sets the Terminal object that this simulation can print data into.

#### Parameters

• ##### terminal: null | Terminal

the Terminal object that this simulation can print data into.

#### Returns void

• Sets the VarsList for this simulation.

#### Parameters

• ##### varsList: VarsList

the VarsList to use in this simulation

#### Returns void

• Called at the start of a mouse drag. The nearest dragable SimObject is passed in, along with mouse position in simulation coordinates. If no dragable SimObject was found, `null` is passed for the first argument. If the EventHandler does not recognize the SimObject then it should return `false`.

#### Parameters

• ##### simObject: null | SimObject

the SimObject that is nearest to the mouse drag coordinates, or `null` if no SimObject was found

• ##### _location: Vector

the location of the mouse in simulation coordinates of the SimView where `simObject` was found, or in the focus view if `simObject` is `null`.

• ##### _offset: Vector

distance from the initial object position (from DisplayObject.getPosition) to the mouse location at start of drag

• ##### _dragBody: null | Vector

location of 'drag point' on the SimObject in body coordinates of the SimObject; this is where for example a spring will be attached on the SimObject when dragging; or `null` when no SimObject was found

• ##### _modifiers: ModifierKeys

the modifier keys down during event

#### Returns boolean

`true` if the EventHandler will handle dragging the SimObject

• Returns a minimal string representation of this object, usually giving just identity information like the class name and name of the object.

For an object whose main purpose is to represent another Printable object, it is recommended to include the result of calling `toStringShort` on that other object. For example, calling `toStringShort()` on a DisplayShape might return something like this:

``````DisplayShape{polygon:Polygon{'chain3'}}
``````

#### Returns string

a minimal string representation of this object.

Generated using TypeDoc