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1 Geometric and kinematic foundations of Lagrangian mechanics |
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1 | (68) |
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1 | (2) |
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1.2 Length of a curve and natural parametrisation |
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3 | (4) |
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1.3 Tangent vector, normal vector and curvature of plane curves |
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7 | (5) |
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12 | (3) |
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1.5 Vector fields and integral curves |
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15 | (1) |
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16 | (17) |
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1.7 Differentiable Riemannian manifolds |
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33 | (13) |
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1.8 Actions of groups and tori |
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46 | (3) |
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1.9 Constrained systems and Lagrangian coordinates |
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49 | (3) |
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52 | (2) |
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54 | (3) |
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1.12 Accelerations of a holonomic system |
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57 | (1) |
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58 | (3) |
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1.14 Additional remarks and bibliographical notes |
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61 | (1) |
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1.15 Additional solved problems |
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62 | (7) |
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2 Dynamics: general laws and the dynamics of a point particle |
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69 | (22) |
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2.1 Revision and comments on the axioms of classical mechanics |
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69 | (2) |
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2.2 The Galilean relativity principle and interaction forces |
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71 | (4) |
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2.3 Work and conservative fields |
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75 | (2) |
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2.4 The dynamics of a point constrained by smooth holonomic constraints |
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77 | (3) |
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2.5 Constraints with friction |
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80 | (1) |
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2.6 Point particle subject to unilateral constraints |
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81 | (2) |
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2.7 Additional remarks and bibliographical notes |
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83 | (1) |
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2.8 Additional solved problems |
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83 | (8) |
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91 | (34) |
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91 | (1) |
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3.2 Analysis of motion due to a positional force |
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92 | (4) |
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96 | (2) |
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3.4 Phase plane and equilibrium |
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98 | (5) |
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3.5 Damped oscillations, forced oscillations. Resonance |
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103 | (4) |
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107 | (1) |
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108 | (4) |
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3.8 Additional remarks and bibliographical notes |
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112 | (1) |
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3.9 Additional solved problems |
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113 | (12) |
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4 The dynamics of discrete systems. Lagrangian formalism |
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125 | (54) |
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125 | (2) |
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4.2 Holonomic systems with smooth constraints |
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127 | (1) |
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128 | (8) |
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4.4 Determination of constraint reactions. Constraints with friction |
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136 | (2) |
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4.5 Conservative systems. Lagrangian function |
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138 | (3) |
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4.6 The equilibrium of holonomic systems with smooth constraints |
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141 | (1) |
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4.7 Generalised potentials. Lagrangian of an electric charge in an electromagnetic field |
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142 | (2) |
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4.8 Motion of a charge in a constant electric or magnetic field |
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144 | (3) |
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4.9 Symmetries and conservation laws. Noether's theorem |
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147 | (3) |
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4.10 Equilibrium, stability and small oscillations |
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150 | (9) |
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159 | (3) |
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162 | (3) |
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4.13 Additional remarks and bibliographical notes |
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165 | (1) |
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4.14 Additional solved problems |
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165 | (14) |
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5 Motion in a central field |
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179 | (34) |
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5.1 Orbits in a central field |
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179 | (6) |
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185 | (2) |
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5.3 Potentials admitting closed orbits |
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187 | (6) |
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193 | (4) |
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197 | (3) |
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200 | (1) |
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201 | (4) |
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205 | (2) |
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5.9 Additional remarks and bibliographical notes |
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207 | (1) |
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5.10 Additional solved problems |
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208 | (5) |
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6 Rigid bodies: geometry and kinematics |
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213 | (22) |
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6.1 Geometric properties. The Euler angles |
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213 | (3) |
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6.2 The kinematics of rigid bodies. The fundamental formula |
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216 | (3) |
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6.3 Instantaneous axis of motion |
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219 | (2) |
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6.4 Phase space of precessions |
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221 | (2) |
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223 | (3) |
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226 | (2) |
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6.7 Ruled surfaces in a rigid motion |
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228 | (2) |
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230 | (1) |
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6.9 Additional solved problems |
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231 | (4) |
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7 The mechanics of rigid bodies: dynamics |
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235 | (44) |
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7.1 Preliminaries: the geometry of masses |
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235 | (1) |
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7.2 Ellipsoid and principal axes of inertia |
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236 | (3) |
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7.3 Homography of inertia |
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239 | (3) |
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7.4 Relevant quantities in the dynamics of rigid bodies |
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242 | (2) |
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7.5 Dynamics of free systems |
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244 | (1) |
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7.6 The dynamics of constrained rigid bodies |
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245 | (5) |
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7.7 The Euler equations for precessions |
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250 | (1) |
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7.8 Precessions by inertia |
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251 | (3) |
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254 | (2) |
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7.10 Integration of Euler equations |
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256 | (3) |
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7.11 Gyroscopic precessions |
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259 | (2) |
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7.12 Precessions of a heavy gyroscope (spinning top) |
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261 | (2) |
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263 | (2) |
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265 | (1) |
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7.15 Additional solved problems |
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266 | (13) |
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8 Analytical mechanics: Hamiltonian formalism |
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279 | (22) |
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8.1 Legendre transformations |
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279 | (3) |
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282 | (2) |
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284 | (1) |
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285 | (2) |
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8.5 Poincaré recursion theorem |
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287 | (1) |
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288 | (3) |
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8.7 Additional remarks and bibliographical notes |
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291 | (1) |
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8.8 Additional solved problems |
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291 | (10) |
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9 Analytical mechanics: variational principles |
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301 | (30) |
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9.1 Introduction to the variational problems of mechanics |
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301 | (1) |
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9.2 The Euler equations for stationary functionals |
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302 | (10) |
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9.3 Hamilton's variational principle: Lagrangian form |
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312 | (2) |
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9.4 Hamilton's variational principle: Hamiltonian form |
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314 | (2) |
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9.5 Principle of the stationary action |
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316 | (2) |
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318 | (5) |
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323 | (1) |
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9.8 Additional remarks and bibliographical notes |
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324 | (1) |
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9.9 Additional solved problems |
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324 | (7) |
| 10 Analytical mechanics: canonical formalism |
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331 | (82) |
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10.1 Symplectic structure of the Hamiltonian phase space |
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331 | (9) |
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10.2 Canonical and completely canonical transformations |
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340 | (12) |
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10.3 The Poincaré-Cartan integral invariant. The Lie condition |
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352 | (12) |
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10.4 Generating functions |
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364 | (7) |
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371 | (3) |
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10.6 Lie derivatives and commutators |
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374 | (6) |
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10.7 Symplectic rectification |
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380 | (4) |
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10.8 Infinitesimal and near-to-identity canonical transformations. Lie series |
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384 | (9) |
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10.9 Symmetries and first integrals |
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393 | (2) |
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10.10 Integral invariants |
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395 | (2) |
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10.11 Symplectic manifolds and Hamiltonian dynamical systems |
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397 | (2) |
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399 | (5) |
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10.13 Additional remarks and bibliographical notes |
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404 | (1) |
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10.14 Additional solved problems |
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405 | (8) |
| 11 Analytic mechanics: Hamilton-Jacobi theory and integrability |
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413 | (74) |
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11.1 The Hamilton-Jacobi equation |
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413 | (8) |
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11.2 Separation of variables for the Hamilton-Jacobi equation |
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421 | (10) |
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11.3 Integrable systems with one degree of freedom: action-angle variables |
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431 | (8) |
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11.4 Integrability by quadratures. Liouville's theorem |
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439 | (7) |
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11.5 Invariant l-dimensional tori. The theorem of Arnold |
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446 | (7) |
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11.6 Integrable systems with several degrees of freedom: action-angle variables |
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453 | (5) |
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11.7 Quasi-periodic motions and functions |
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458 | (8) |
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11.8 Action-angle variables for the Kepler problem. Canonical elements, Delaunay and Poincaré variables |
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466 | (5) |
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11.9 Wave interpretation of mechanics |
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471 | (6) |
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477 | (3) |
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11.11 Additional remarks and bibliographical notes |
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480 | (1) |
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11.12 Additional solved problems |
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481 | (6) |
| 12 Analytical mechanics: canonical perturbation theory |
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487 | (58) |
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12.1 Introduction to canonical perturbation theory |
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487 | (12) |
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12.2 Time periodic perturbations of one-dimensional uniform motions |
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499 | (3) |
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12.3 The equation Dωu = upsilon Conclusion of the previous analysis |
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502 | (5) |
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12.4 Discussion of the fundamental equation of canonical perturbation theory. Theorem of Poincaré on the non-existence of first integrals of the motion |
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507 | (9) |
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12.5 Birkhoff series: perturbations of harmonic oscillators |
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516 | (6) |
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12.6 The Kolmogorov-Arnol'd-Moser theorem |
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522 | (7) |
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12.7 Adiabatic invariants |
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529 | (3) |
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532 | (2) |
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12.9 Additional remarks and bibliographical notes |
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534 | (1) |
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12.10 Additional solved problems |
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535 | (10) |
| 13 Analytical mechanics: an introduction to ergodic theory and to chaotic motion |
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545 | (46) |
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13.1 The concept of measure |
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545 | (3) |
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13.2 Measurable functions. Integrability |
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548 | (2) |
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13.3 Measurable dynamical systems |
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550 | (4) |
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13.4 Ergodicity and frequency of visits |
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554 | (9) |
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563 | (2) |
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565 | (6) |
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13.7 Computation of the entropy. Bernoulli schemes. Isomorphism of dynamical systems |
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571 | (4) |
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13.8 Dispersive billiards |
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575 | (3) |
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13.9 Characteristic exponents of Lyapunov. The theorem of Oseledec |
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578 | (3) |
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13.10 Characteristic exponents and entropy |
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581 | (1) |
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13.11 Chaotic behaviour of the orbits of planets in the Solar System |
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582 | (2) |
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584 | (2) |
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13.13 Additional solved problems |
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586 | (4) |
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13.14 Additional remarks and bibliographical notes |
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590 | (1) |
| 14 Statistical mechanics: kinetic theory |
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591 | (22) |
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14.1 Distribution functions |
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591 | (1) |
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14.2 The Boltzmann equation |
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592 | (4) |
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14.3 The hard spheres model |
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596 | (3) |
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14.4 The Maxwell-Boltzmann distribution |
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599 | (2) |
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14.5 Absolute pressure and absolute temperature in an ideal monatomic gas |
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601 | (3) |
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604 | (1) |
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14.7 The 'H theorem' of Boltzmann. Entropy |
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605 | (4) |
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609 | (1) |
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14.9 Additional solved problems |
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610 | (1) |
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14.10 Additional remarks and bibliographical notes |
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611 | (2) |
| 15 Statistical mechanics: Gibbs sets |
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613 | (58) |
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15.1 The concept of a statistical set |
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613 | (3) |
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15.2 The ergodic hypothesis: averages and measurements of observable quantities |
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616 | (4) |
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15.3 Fluctuations around the average |
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620 | (1) |
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15.4 The ergodic problem and the existence of first integrals |
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621 | (3) |
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15.5 Closed isolated systems (prescribed energy). Micro canonical set |
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624 | (3) |
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15.6 Maxwell—Boltzmann distribution and fluctuations in the microcanonical set |
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627 | (4) |
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631 | (3) |
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15.8 Equipartition of the energy (prescribed total energy) |
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634 | (2) |
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15.9 Closed systems with prescribed temperature. Canonical set |
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636 | (4) |
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15.10 Equipartition of the energy (prescribed temperature) |
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640 | (5) |
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15.11 Helmholtz free energy and orthodicity of the canonical set |
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645 | (1) |
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15.12 Canonical set and energy fluctuations |
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646 | (1) |
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15.13 Open systems with fixed temperature. Grand canonical set |
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647 | (4) |
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15.14 Thermodynamical limit. Fluctuations in the grand canonical set |
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651 | (3) |
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654 | (2) |
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656 | (3) |
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15.17 Additional remarks and bibliographical notes |
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659 | (3) |
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15.18 Additional solved problems |
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662 | (9) |
| 16 Lagrangian formalism in continuum mechanics |
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671 | (24) |
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16.1 Brief summary of the fundamental laws of continuum mechanics |
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671 | (5) |
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16.2 The passage from the discrete to the continuous model. The Lagrangian function |
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676 | (2) |
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16.3 Lagrangian formulation of continuum mechanics |
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678 | (2) |
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16.4 Applications of the Lagrangian formalism to continuum mechanics |
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680 | (4) |
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16.5 Hamiltonian formalism |
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684 | (1) |
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16.6 The equilibrium of continua, as a variational problem. Suspended cables |
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685 | (5) |
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690 | (1) |
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16.8 Additional solved problems |
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691 | (4) |
| Appendices |
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Appendix 1: Some basic results on ordinary differential equations |
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695 | (10) |
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695 | (2) |
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A1.2 Systems of equations with constant coefficients |
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697 | (4) |
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A1.3 Dynamical systems on manifolds |
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701 | (4) |
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Appendix 2: Elliptic integrals and elliptic functions |
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705 | (4) |
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Appendix 3: Second fundamental form of a surface |
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709 | (6) |
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Appendix 4: Algebraic forms, differential forms, tensors |
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715 | (14) |
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715 | (4) |
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719 | (5) |
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724 | (2) |
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726 | (3) |
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Appendix 5: Physical realisation of constraints |
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729 | (4) |
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Appendix 6: Kepler's problem, linear oscillators and geodesic flows |
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733 | (8) |
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Appendix 7: Fourier series expansions |
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741 | (4) |
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Appendix 8: Moments of the Gaussian distribution and the Euler Γ function |
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745 | (4) |
| Bibliography |
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749 | (10) |
| Index |
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759 | |
Other versions by this Author
