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9ème (bis) Cours de Mécanique Analytique II (2016-17), Bac3 phys, Master 1 Sc. Sp.

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3 • 2.0 Transformée de Legendre : notions

L(v’) v’ v Si p = p(v) = (dL/dv’)v y(v,v’) = p(v) (v’-v) + L(v) y(v,0) y(v,0) = -p(v) v + L(v) Si dp(v)/dv ≠ 0; d2L/d2v ≠ 0 ⇒ v = v(p) Lv→p(L(v)) = H(p) = -y(v,0) = p v(p) - L(v(p)) =-H(p)

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• 2.0 Transformée de Legendre : notions

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5 • 2.0 Transformée de Legendre : notions

v y(v,0) y(v,0) = -p(v) v + L(v) =-H(p) L(v) Lv→p(L(v)) = H(p) = -y(v,0) = p v(p) - L(v(p))

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7 • 2.0 Transformée de Legendre : notions

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• 2.1 Equations de Hamilton et systèmes canoniques Lagrangien : f; qi(t), ...⇒ Hamiltonien : 2f; qi(t), pi(t), t (2.1) (2.3) (2.2) (2.4) (2.5)

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9 • 2.1 Equations de Hamilton et systèmes canoniques

(2.6)

(2.7) (2.8) (2.9)

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• 2.1 Equations de Hamilton et systèmes canoniques (2.6) (2.9) (2.10) Si L(q1, q2, .. qj-1, qj, qj+1, … qf, q, t) ⇒ H(q1, q2, .. qj-1, qj, qj+1, … qf, p, t) Si L(q, q, t) ⇒ H(q, p, t) • •

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11 • 2.1 Equations de Hamilton et systèmes canoniques

Suivant Lagrange ... Espace de configuration q1 qj qj+1 qf • q1 qj qj+1 qf • Suivant Hamilton ... p1 pf pj pj+1 Espace de phase

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13 • 2.2 Exemples de systèmes canoniques

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15 • 2.2 Exemples de systèmes canoniques

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17 • 2.2 Exemples de systèmes canoniques

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19 • 2.2 Exemples de systèmes canoniques

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21 • 2.3 Le principe variationnel d’Hamilton modifié

(2.11) (2.12)

(2.13) (2.14)

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