"Action" from Data Points

Since we need to calculate a definite integral using a finite number of points, we should start by discretising: $\int\limits_{t_1}^{t_2} (E-U) \textrm{ d}t \approx \sum_{i} (E_i - U_i) \Delta t =\sum_{i} E_i \Delta t -\sum_{i} U_i\Delta t$

We can take $\Delta t = 0.001\textrm{ s}$ , since it was the time step used throughout the data file.

Now we need to calculate the kinetic energy, which is defined by $E_i = \frac{1}{2}m|\vec{v}|^2$ , where $\vec{v}$ is the velocity. Since we are dealing with a 2-dimensional problem, we can decompose this vector in its $x$ - and $y$ -components, so that: $|\vec{v}|^2=v_x^2+v_y^2,$ where $v_x = \dfrac{\textrm{d}x}{\textrm{d}t}$ and $v_y = \dfrac{\textrm{d}y}{\textrm{d}t}$

We can now approximate these derivatives using finite differences: $\dfrac{\textrm{d}x}{\textrm{d}t}(t_i)\approx \dfrac{x(t_i+\Delta t) - x(t_i)}{\Delta t}$

Proceeding analogously for $v_y$ , the kinetic energy yields: $E_i \approx \dfrac{1}{2}m\left[\left(\dfrac{x(t_i+\Delta t) - x(t_i)}{\Delta t}\right)^2+\left(\dfrac{y(t_i+\Delta t) - y(t_i)}{\Delta t}\right)^2\right]$

And its time-integral is: $\sum_{i} E_i \Delta t = \sum_{i} \dfrac{1}{2}m\left[\left(\dfrac{x(t_i+\Delta t) - x(t_i)}{\Delta t}\right)^2+\left(\dfrac{y(t_i+\Delta t) - y(t_i)}{\Delta t}\right)^2\right] \Delta t = \dfrac{m}{2\Delta t}\sum_{i}\left[\left(x(t_i+\Delta t) - x(t_i)\right)^2+\left(y(t_i+\Delta t) - y(t_i)\right)^2\right]$

The potential energy is simply: $U_i = mgy(t_i)$

Hence, its time-integral is: $\sum_{i} U_i \Delta t = mg\Delta t\sum_{i} y(t_i)$

We are now able to write and run a small Python script to perform the calculations for us:

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#Some variables and initialisations
m = 1 #mass (kg)
g = 10 #gravity (m/s^2)
dt = 0.001 #time step (s)
length = 0 #line counter
data = [] #empty list to contain the data read from the file

#Opens the input file
with open('action_calc_data.txt') as input:  
    #Ignores the first line, since it is only a header containing the column labels
    next(input)
    #Runs a cycle to go through each line of the file
    for line in input:
        #Counts the number of lines, adding 1 to the 'length' counter; we need this number in order to bound our loops below
        length+=1
        #Splits each line of the file, using white spaces as separators, and appends it to the 'data' list
        data.append(line.split())

#Calculates the potential energy integral, performing a summation along the y column (whose index is 2)
integral_U = m*g*dt*sum(float(data[i][2]) for i in range(length))

#Calculates the kinetic energy integral, performing a summation along both the x and y columns (whose indexes are 1 and 2, respectively)
#Notice that the range argument must be length-1 instead of length; otherwise, when j=length-1, we would be accessing:
#data[length+1]=data[length+1-1]=data[length], which is out of range!
integral_E = .5*m/dt*sum((float(data[j+1][1])-float(data[j][1]))**2.0 + (float(data[j+1][2])-float(data[j][2]))**2.0 for j in range(length-1)) 

#Simply prints out our answer, which is the difference between the two integrals
print('The action is:\nS = ' + str(integral_E-integral_U) + ' J s') 

The action is:
S = -35.76439363705195 J s

So the answer to 3 significant figures is: $S \approx -35.8 \textrm{ J s}$