Preface |
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ix | |
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1 Elements of the Time Scale Calculus |
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1 | (78) |
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1.1 Forward and Backward Jump Operators, Graininess Function |
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1 | (4) |
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5 | (7) |
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12 | (7) |
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1.4 The Exponential Function |
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19 | (10) |
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1.5 Hyperbolic and Trigonometric Functions |
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29 | (1) |
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1.6 The Multidimensional Time Scale Calculus |
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30 | (36) |
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66 | (6) |
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72 | (3) |
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1.9 Advanced Practical Problems |
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75 | (4) |
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2 Dynamic Systems on Time Scales |
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79 | (52) |
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2.1 Structure of Dynamic Systems on Time Scales |
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79 | (36) |
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2.2 Constant Coefficients |
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115 | (12) |
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2.3 Advanced Practical Problems |
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127 | (4) |
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131 | (26) |
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3.1 Definition for Functionals |
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131 | (2) |
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3.2 Self-Adjoint Second Order Matrix Equations |
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133 | (11) |
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144 | (9) |
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153 | (4) |
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4 Linear Hamiltonian Dynamic Systems |
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157 | (62) |
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4.1 Linear Symplectic Dynamic Systems |
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157 | (6) |
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163 | (3) |
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166 | (11) |
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177 | (5) |
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182 | (16) |
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4.6 "Big" Linear Hamiltonian Systems |
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198 | (13) |
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4.7 Positivity of Quadratic Functionals |
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211 | (8) |
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219 | (32) |
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5.1 The Dubois-Reymond Lemma |
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219 | (6) |
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5.2 The Variational Problem |
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225 | (8) |
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5.3 The Euler-Lagrange Equation |
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233 | (8) |
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241 | (4) |
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245 | (3) |
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5.6 Advanced Practical Problems |
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248 | (3) |
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6 Higher Order Calculus of Variations |
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251 | (10) |
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6.1 Statement of the Variational Problem |
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251 | (3) |
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254 | (6) |
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6.3 Advanced Practical Problems |
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260 | (1) |
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7 Double Integral Calculus of Variations |
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261 | (14) |
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7.1 Statement of the Variational Problem |
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262 | (1) |
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7.2 First and Second Variation |
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263 | (5) |
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268 | (4) |
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7.4 Advanced Practical Problems |
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272 | (3) |
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8 Noether's Second Theorem |
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275 | (16) |
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8.1 Invariance under Transformations |
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275 | (4) |
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8.2 Noether's Second Theorem without Transformations of Time |
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279 | (2) |
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8.3 Noether's Second Theorem with Transformations of Time |
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281 | (4) |
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8.4 Noether's Second Theorem-Double Delta Integral Case |
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285 | (6) |
References |
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291 | (2) |
Author Contact Information |
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293 | (2) |
Index |
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295 | |