If you visit the page I have just created, you may find the simulation of a mechanical system.

It is currently $$\mathcal H=p_1^2+p_2^2-\cos q_1-\cos q_2- \cos\!\left(q_1-q_2\right)$$ depicting two pendulum coupled with a spring (with $l_1=l_2$ and $m_1=m_2$, and the original length of the spring is zero, and the two hanging points overlap),

which is a classical example of non-linearly coupled system.

The pattern of the oscillation can be analyzed using discrete Fourier transformation, whose result can be found by clicking the buttons in the up-left corner (after the simulator has detected a period).

Hitting the space bar can make the simulation pause.

If you want to use it to simulate other mechanical systems, you can study the codes I wrote and write your own codes in the console.

By the way, the OpenRGSS version of the simulator is open-source here. Please star the repo if you like it.

To utilize the canonical equations $$\frac{\mathrm d\mathbf q}{\mathrm dt}= \frac{\partial\mathcal H}{\partial\mathbf p},\quad \frac{\mathrm d\mathbf p}{\mathrm dt}= -\frac{\partial\mathcal H}{\partial\mathbf p},$$ $$(1)$$ we need to calculate the partial derivatives of $\mathcal H$. Here is a simple code to calculate partial derivatives.

          
            def div x0, dx, f

            	f0 = f.(x0)

            	n = x0.size

            	Array.new n do |i|

            		(f.(x0 + Vector.basis(n, i) * dx) - f0) / dx

            	end

            end

          
        

(RGSS do not have matrix.rb, you can copy one from the attached file below.) Here x0 is a Vector, f is a call-able object as a function of vectors, dx is a small scalar which we are going to take 1e-6.

Let x = Vector[*q, *p], and then Formula 1 has the form $$\frac{\mathrm d\mathbf x}{\mathrm dt}=f\!\left(\mathbf x\right).$$ To solve this equation numerically, we need to use a famous method called the (explicit) Runge–Kutta method.

          
            def runge_kutta initial, max_t, dt, (*pyramid, coefs), func

            	(0..max_t).step(dt).reduce initial do |ret, t|

            		$canvas.trace t, ret if $canvas

            		coefs.zip(pyramid).each_with_object([]).sum do |(coef, row), ary|

            			coef * ary.push(func.(t, row.inner(ary) * dt + ret)).last

            		end * dt + ret#p(ret)

            	end

            end

          
        

Note that Runge–Kutta is a family of methods. The argument (*pyramid, coefs) takes one of the following, each of which is a single Runge–Kutta method.

          
            FORWARD_EULER = [[],[1]]

            EXPLICIT_MIDPOINT = [[],[1/2.0],[0,1]]

            HEUN = [[],[1],[1/2.0,1/2.0]]

            RALSTON = [[],[2/3.0],[1/4.0,3/4.0]]

            KUTTA_3RD = [[],[1/2.0],[-1,2],[1/6.0,2/3.0,1/6.0]]

            HEUN_3RD = [[],[1/3.0],[0,2/3.0],[1/4.0,0,3/4.0]]

            RALSTON_3RD = [[],[1/2.0],[0,3/4.0],[2/9.0,1/3.0,4/9.0]]

            SSPRK3 = [[],[1],[1/4.0,1/4.0],[1/6.0,1/6.0,2/3.0]]

            CLASSIC_4TH = [[],[1/2.0],[0,1/2.0],[0,0,1],[1/6.0,1/3.0,1/3.0,1/6.0]]

            RALSTON_4TH = [[],[0.4],[0.29697761,0.15875964],[0.21810040,-3.05096516,

            		3.83286476],[0.17476028, -0.55148066, 1.20553560, 0.17118478]]

            THREE_EIGHTH_4TH = [[],[1/3.0],[-1/3.0,1],[1,-1,1],[1/8.0,3/8.0,3/8.0,1/8.0]]

          
        

Here we are going to take CLASSIC_4TH.

The $canvas appearing here is an object that is going to draw the result onto the screen.

Here we also need to have some patches to get it work.

          
            class Float

            	alias ulysses20200426121236_add +

            	def + other

            		if zero? && [Vector, Matrix].any? { |c| other.is_a? c }

            			other

            		else

            			ulysses20200426121236_add other

            		end

            	end

            end

            module Enumerable

            	def sum init = 0, &block

            		(block ? map(&block) : self).reduce init, :+

            	end

            end

            class Array

            	def inner other

            		zip(other).sum { |a, b| a * b }

            	end

            end

          
        

(Again, note that this is Ruby 1.9.2.)

Finally, just combine them up, and we can solve a Hamiltonian numerically.

          
            def solve_hamiltonian n, qp0, max_t, dt, hamiltonian

            	runge_kutta qp0, max_t, dt, CLASSIC_4TH, ->t, qp do

            		dqpdt = div qp, 1e-6, ->x { hamiltonian.(t, x) }

            		Vector[*dqpdt[n...n*2], *dqpdt[0...n].map(&:-@)]

            	end

            end

          
        

For example, let’s simulate a double pendulum.

          
            solve_hamiltonian 2,Vector[PI/2,0.0,0.0,0.0],Float::INFINITY,1e-3,

            		->t,(q1,q2,p1,p2){p1**2+p2**2/2+cos(q1-q2)*p1*p2-cos(q1)-cos(q2)}

          
        

The codes are not complete in this post. See the attached file for details. You can open the project using RPG Maker VX Ace. The Game.exe file is not the official Game.exe executable but the third-party improved version of it called RGD (of version 1.3.2, while the latest till now is 1.5.1).

I only have experience in using RPG Maker XP, RPG Maker VX, RPG Maker VX Ace, and RPG Maker MV.

Games made by using RM are based on

In most cases, when I say RGSS, I mean RGSS, RGSS2, and RGSS3. RGSS is in Ruby and rpg_core.js is in Javascript.

The scripts of a basic RGSS game look like

          
            def start

            	# some codes

            end

            def update

            	# some codes

            end

            

            start

            loop do

            	update

            	Input.update

            	Graphics.update

            end

          
        

The scripts of a basic rpg_core.js game look like

          
            function start() {

            	// some codes

            }

            function update() {

            	// some codes

            }

            

            Graphics.initialize(816, 624, 'webgl');

            Graphics.boxWidth = 816;

            Graphics.boxHeight = 624;

            WebAudio.initialize(false);

            Input.initialize();

            TouchInput.initialize();

            var deltaTime = 1.0 / 60.0;

            var accumulator = 0.0;

            var currentTime;

            window.scene = new Stage();

            start();

            function performUpdate() {

            	Graphics.tickStart();

            	var newTime = performance.now();

            	if (currentTime === undefined) currentTime = newTime;

            	var fTime = ((newTime - currentTime) / 1000).clamp(0, 0.25);

            	currentTime = newTime;

            	accumulator += fTime;

            	while (accumulator >= deltaTime) {

            		Input.update();

            		TouchInput.update();

            		update();

            		accumulator -= deltaTime;

            	}

            	Graphics.render(window.scene);

            	requestAnimationFrame(performUpdate);

            	Graphics.tickEnd();

            }

            performUpdate();

          
        

There is a simple rpg_core.js game written by me which can be accessed here. You can look at its source code by using your browser.

RM also provides a lot of scripts serving as making RPG. There is a simple RPG built by me which can be accessed here.

The source codes of RGSS are secret, but those of rpg_core.js are not a secret and can be seen at its GitHub repo.