Second Order Hamiltonian Systems

Martin Schechter

Chapter Chapter 10 in Critical Point Theory, 2020, pp 167-190 from Springer

Abstract: Abstract We consider the system − x ̈ ( t ) = B ( t ) x ( t ) + ∇ x V ( t , x ( t ) ) , $$\displaystyle -\ddot x(t)=\; B(t)x(t) +\nabla _xV(t,x(t)), $$ where x ( t ) = ( x 1 ( t ) , ⋯ , x n ( t ) ) $$\displaystyle x(t)=(x_1(t),\cdots ,x_n(t)) $$ is a map from I = [0, T] to ℝ n $$\mathbb R^n$$ such that each component x j(t) is a periodic function in H 1 with period T, and the function V (t, x) = V (t, x 1, ⋯ , x n) is continuous from ℝ n + 1 $$\mathbb R^{n+1}$$ to ℝ $$\mathbb R$$ with ∇ x V ( t , x ) = ( ∂ V ∕ ∂ x 1 , ⋯ , ∂ V ∕ ∂ x n ) ∈ C ( ℝ n + 1 , ℝ n ) . $$\displaystyle \nabla _xV(t,x)=(\partial V/\partial x_1,\cdots ,\partial V/ \partial x_n) \in C(\mathbb R^{n+1},\mathbb R^n). $$ For each x ∈ ℝ n , $$x \in \mathbb R^n,$$ the function V (t, x) is periodic in t with period T.

Date: 2020
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DOI: 10.1007/978-3-030-45603-0_10

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