By Ovidiu Calin, Der-Chen Chang
* a geometrical method of difficulties in physics, lots of which can't be solved through the other methods
* textual content is enriched with reliable examples and workouts on the finish of each chapter
* high quality for a path or seminar directed at grad and adv. undergrad scholars attracted to elliptic and hyperbolic differential equations, differential geometry, calculus of adaptations, quantum mechanics, and physics
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Extra info for Geometric mechanics on Riemannian manifolds : applications to partial differential equations
Example text
15) Proof. Applying the chain rule dI du As δ ( i) ;e = (δ | u=0 i) ∂L δ i D ∂ = ; e, i + ∂L i ) ;e ∂( i ; e) δ( dvg . the second term in the right hand side can be expressed as ∂L D i i ∂( i ) ;e i δ ;e − Let X = Xe where ∂L i ∂( δ i dvg . ∂ , ∂xe ∂L Xe = i ) ;e ;e ∂( i ) ;e δ i , and by the divergence theorem D X e; e dv = 0, as X vanishes on ∂D. 15) is satisfied. Indeed, if we take the variation (s, x) = exp(s V (x) ), where V (x) ∈ T (x) N, we have (0, x) = (x) and ∂ (s, x) δ = = V (x) , ∂s | s=0 for any arbitrary V .
22) Proof. 19) with the substitution X = ∇φ. 2 Applications Harmonic functions on compact manifolds The compact manifold M considered in this section will have an empty boundary ∂M = ∅. 15. ( Hopf’s lemma) Let M be a connected, compact Riemannian manifold and f ∈ F(M) such that f ≥ 0. Then f is constant. Proof. First, we shall show that f =0 on M. This is obtained by integrating and applying the divergence theorem 0≤ div(∇f ) dv = 0, f dv = − M M where we used ∂M = 0. 22), we get div(f ∇f ) = −f f + g(∇f, ∇f ).
8) The dynamical system discussed above is one dimensional. However, it was not easy to integrate the Euler–Lagrange equation, even in the particular case C = 0. The solution required the use of elliptic functions. In other cases, even elliptic functions are not enough to solve the Euler–Lagrange equation. We may say that for some equations, it is not possible to obtain explicit formulas. This is also the case for an Euler–Lagrange equation on manifolds, where we encounter more than one parameter.