0: #!/usr/bin/perl
1: use strict;
2: use warnings;
3: 
4: # Perl solving a physics / electrodynamics problem involving
5: # symbolic mathematics, derivatives and complex numbers:
6: 
7: use Math::Symbolic qw/:all/;
8: use Math::Complex;
9: 
10: # Given the following simple circuit:
11: #
12: #  ----|||||-----/\/\/\----       (R = resistor,
13: # |      R          L      |       L = solenoid,
14: # |                        |       U = alternating voltage)
15: #  ---------O ~ O----------
16: #            U(t)
17: #
18: # Question: What's the current in this circuit?
19: #
20: # We'll need some physics before letting the computer do the
21: # math:
22: # Applying Kirchhoff's rules, one quickly ends up with the
23: # following differential equation for the current:
24: #     (L * dI/dt) + (R * I)  =  U
25: 
26: my $left  = parse_from_string('L * total_derivative(I(t), t) + R * I(t)');
27: my $right = parse_from_string('U(t)');
28: 
29: <readmore>
30: # If we understand current and voltage to be complex functions,
31: # we'll be able to derive. ("'" denoting complex here)
32: #    I'(t) = I'_max * e^(i*omega*t)
33: #    U'(t) = U_max  * e^(i*omega*t)
34: # (Please note that omega is the frequency of the alternating voltage.
35: # For example, the voltage from German outlets has a frequency of 50Hz.)
36: 
37: my $argument = parse_from_string('e^(i*omega*t)');
38: my $current  = parse_from_string('I_max') * $argument;
39: my $voltage  = parse_from_string('U_max') * $argument;
40: 
41: # Putting it into the equation:
42: $left->implement( I  => $current );
43: $right->implement( U => $voltage );
44: 
45: $left = $left->apply_derivatives()->simplify();
46: 
47: # Now, we can solve the equation to get a complex function for
48: # the current:
49: 
50: $left  /= $argument;
51: $right /= $argument;
52: my $quotient = parse_from_string('R + i*omega*L');
53: $left  /= $quotient;
54: $right /= $quotient;
55: 
56: # Now we have:
57: #    $left    = $right
58: #    I_max(t) = U_max / (R + i*omega*L)
59: # But I_max(t) is still complex and so is the right-hand-side of the
60: # equation!
61: 
62: # Making the symbolic i a "literal" Math::Complex i
63: $right->implement(
64:     e => Math::Symbolic::Constant->euler(),
65:     i => Math::Symbolic::Constant->new(i),    # Math::Complex magic
66: );
67: 
68: print <<'HERE';
69: Sample of complex maximum current with the following values:
70:   U_max => 100
71:   R     => 10
72:   L     => 10
73:   omega => 1
74: HERE
75: 
76: print "Computed to: "
77:   . $right->value(
78:     U_max => 100,
79:     R     => 10,
80:     L     => 10,
81:     omega => 1,
82:   ),
83:   "\n\n";
84: 
85: # Now, we're dealing with alternating current and voltage.
86: # So let's make a generator that generates nice current
87: # functions of time!
88: #   I(t) = Re(I_max(t)) * cos(omega*t - phase);
89: 
90: # Usage: generate_current(U_Max, R, L, omega, phase)
91: sub generate_current {
92:     my $current = $right->new();    # cloning
93: 
94:     $current *= parse_from_string('cos(omega*t - phase)');
95: 
96:     $current->implement(
97:         U_max => $_[0],
98:         R     => $_[1],
99:         L     => $_[2],
100:         omega => $_[3],
101:         phase => $_[4],
102:     );
103:     $current = $current->simplify();
104:     return sub { Re( $current->value( t => $_[0] ) ) };
105: }
106: 
107: print "Sample current function with: 230V, 2Ohms, 0.1H, 50Hz, PI/4\n";
108: my $current_of_time = generate_current( 230, 2, 0.1, 50, PI / 4 );
109: 
110: print "The current at 0 seconds:   " . $current_of_time->(0) . "\n";
111: print "The current at 0.1 seconds: " . $current_of_time->(0.1) . "\n";
112: print "The current at 1 second:    " . $current_of_time->(1) . "\n";
113: 
114: </readmore>