1 /** @file exam_indexed.cpp
3 * Here we test manipulations on GiNaC's indexed objects. */
6 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
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16 * GNU General Public License for more details.
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25 static unsigned check_equal(const ex &e1, const ex &e2)
29 clog << e1 << "-" << e2 << " erroneously returned "
30 << e << " instead of 0" << endl;
36 static unsigned check_equal_simplify(const ex &e1, const ex &e2)
38 ex e = simplify_indexed(e1) - e2;
40 clog << "simplify_indexed(" << e1 << ")-" << e2 << " erroneously returned "
41 << e << " instead of 0" << endl;
47 static unsigned delta_check(void)
49 // checks identities of the delta tensor
53 symbol s_i("i"), s_j("j"), s_k("k");
54 idx i(s_i, 3), j(s_j, 3), k(s_k, 3);
58 result += check_equal(delta_tensor(i, j), delta_tensor(j, i));
60 // trace = dimension of index space
61 result += check_equal(delta_tensor(i, i), 3);
62 result += check_equal_simplify(delta_tensor(i, j) * delta_tensor(i, j), 3);
64 // contraction with delta tensor
65 result += check_equal_simplify(delta_tensor(i, j) * indexed(A, k), delta_tensor(i, j) * indexed(A, k));
66 result += check_equal_simplify(delta_tensor(i, j) * indexed(A, j), indexed(A, i));
67 result += check_equal_simplify(delta_tensor(i, j) * indexed(A, i), indexed(A, j));
68 result += check_equal_simplify(delta_tensor(i, j) * delta_tensor(j, k) * indexed(A, i), indexed(A, k));
73 static unsigned metric_check(void)
75 // checks identities of the metric tensor
79 symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma");
80 varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4);
83 // becomes delta tensor if indices have opposite variance
84 result += check_equal(metric_tensor(mu, nu.toggle_variance()), delta_tensor(mu, nu.toggle_variance()));
86 // scalar contraction = dimension of index space
87 result += check_equal(metric_tensor(mu, mu.toggle_variance()), 4);
88 result += check_equal_simplify(metric_tensor(mu, nu) * metric_tensor(mu.toggle_variance(), nu.toggle_variance()), 4);
90 // contraction with metric tensor
91 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu), metric_tensor(mu, nu) * indexed(A, nu));
92 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu.toggle_variance()), indexed(A, mu));
93 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, mu.toggle_variance()), indexed(A, nu));
94 result += check_equal_simplify(metric_tensor(mu, nu) * metric_tensor(mu.toggle_variance(), rho.toggle_variance()) * indexed(A, nu.toggle_variance()), indexed(A, rho.toggle_variance()));
95 result += check_equal_simplify(metric_tensor(mu, rho) * metric_tensor(nu, sigma) * indexed(A, rho.toggle_variance(), sigma.toggle_variance()), indexed(A, mu, nu));
96 result += check_equal_simplify(indexed(A, mu.toggle_variance()) * metric_tensor(mu, nu) - indexed(A, mu.toggle_variance()) * metric_tensor(nu, mu), 0);
97 result += check_equal_simplify(indexed(A, mu.toggle_variance(), nu.toggle_variance()) * metric_tensor(nu, rho), indexed(A, mu.toggle_variance(), rho));
99 // contraction with delta tensor yields a metric tensor
100 result += check_equal_simplify(delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho), metric_tensor(mu, rho));
101 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu.toggle_variance()) * delta_tensor(mu.toggle_variance(), rho), indexed(A, rho));
106 static unsigned epsilon_check(void)
108 // checks identities of the epsilon tensor
112 symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma");
113 varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4);
116 result += check_equal(lorentz_eps(mu, nu, rho, sigma) + lorentz_eps(sigma, rho, mu, nu), 0);
118 // convolution is zero
119 result += check_equal(lorentz_eps(mu, nu, rho, nu.toggle_variance()), 0);
120 result += check_equal(lorentz_eps(mu, nu, mu.toggle_variance(), nu.toggle_variance()), 0);
121 result += check_equal_simplify(lorentz_g(mu.toggle_variance(), nu.toggle_variance()) * lorentz_eps(mu, nu, rho, sigma), 0);
126 static unsigned symmetry_check(void)
128 // check symmetric/antisymmetric objects
132 symbol s_i("i"), s_j("j"), s_k("k");
133 idx i(s_i, 3), j(s_j, 3), k(s_k, 3);
137 result += check_equal(indexed(A, indexed::symmetric, i, j), indexed(A, indexed::symmetric, j, i));
138 result += check_equal(indexed(A, indexed::antisymmetric, i, j) + indexed(A, indexed::antisymmetric, j, i), 0);
139 result += check_equal(indexed(A, indexed::antisymmetric, i, j, k) - indexed(A, indexed::antisymmetric, j, k, i), 0);
144 static unsigned edyn_check(void)
146 // Relativistic electrodynamics
148 // Test 1: check transformation laws of electric and magnetic fields by
149 // applying a Lorentz boost to the field tensor
154 ex gamma = 1 / sqrt(1 - pow(beta, 2));
155 symbol Ex("Ex"), Ey("Ey"), Ez("Ez");
156 symbol Bx("Bx"), By("By"), Bz("Bz");
158 // Lorentz transformation matrix (boost along x axis)
161 L.set(0, 1, -beta*gamma);
162 L.set(1, 0, -beta*gamma);
167 // Electromagnetic field tensor
183 symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma");
184 varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4);
186 // Apply transformation law of second rank tensor
187 ex e = (indexed(L, mu, rho.toggle_variance())
188 * indexed(L, nu, sigma.toggle_variance())
189 * indexed(F, rho, sigma)).simplify_indexed();
191 // Extract transformed electric and magnetic fields
192 ex Ex_p = e.subs(lst(mu == 1, nu == 0)).normal();
193 ex Ey_p = e.subs(lst(mu == 2, nu == 0)).normal();
194 ex Ez_p = e.subs(lst(mu == 3, nu == 0)).normal();
195 ex Bx_p = e.subs(lst(mu == 3, nu == 2)).normal();
196 ex By_p = e.subs(lst(mu == 1, nu == 3)).normal();
197 ex Bz_p = e.subs(lst(mu == 2, nu == 1)).normal();
200 result += check_equal(Ex_p, Ex);
201 result += check_equal(Ey_p, gamma * (Ey - beta * Bz));
202 result += check_equal(Ez_p, gamma * (Ez + beta * By));
203 result += check_equal(Bx_p, Bx);
204 result += check_equal(By_p, gamma * (By + beta * Ez));
205 result += check_equal(Bz_p, gamma * (Bz - beta * Ey));
207 // Test 2: check energy density and Poynting vector of electromagnetic field
216 // Covariant field tensor
217 ex F_mu_nu = (indexed(eta, mu.toggle_variance(), rho.toggle_variance())
218 * indexed(eta, nu.toggle_variance(), sigma.toggle_variance())
219 * indexed(F, rho, sigma)).simplify_indexed();
221 // Energy-momentum tensor
222 ex T = (-indexed(eta, rho, sigma) * F_mu_nu.subs(s_nu == s_rho)
223 * F_mu_nu.subs(lst(s_mu == s_nu, s_nu == s_sigma))
224 + indexed(eta, mu.toggle_variance(), nu.toggle_variance())
225 * F_mu_nu.subs(lst(s_mu == s_rho, s_nu == s_sigma))
226 * indexed(F, rho, sigma) / 4).simplify_indexed() / (4 * Pi);
228 // Extract energy density and Poynting vector
229 ex E = T.subs(lst(s_mu == 0, s_nu == 0)).normal();
230 ex Px = T.subs(lst(s_mu == 0, s_nu == 1));
231 ex Py = T.subs(lst(s_mu == 0, s_nu == 2));
232 ex Pz = T.subs(lst(s_mu == 0, s_nu == 3));
235 result += check_equal(E, (Ex*Ex+Ey*Ey+Ez*Ez+Bx*Bx+By*By+Bz*Bz) / (8 * Pi));
236 result += check_equal(Px, (Ez*By-Ey*Bz) / (4 * Pi));
237 result += check_equal(Py, (Ex*Bz-Ez*Bx) / (4 * Pi));
238 result += check_equal(Pz, (Ey*Bx-Ex*By) / (4 * Pi));
243 unsigned exam_indexed(void)
247 cout << "examining indexed objects" << flush;
248 clog << "----------indexed objects:" << endl;
250 result += delta_check(); cout << '.' << flush;
251 result += metric_check(); cout << '.' << flush;
252 result += epsilon_check(); cout << '.' << flush;
253 result += symmetry_check(); cout << '.' << flush;
254 result += edyn_check(); cout << '.' << flush;
257 cout << " passed " << endl;
258 clog << "(no output)" << endl;
260 cout << " failed " << endl;