  We prove that on the twisted cotangent bundle of a closed manifold with an
exact magnetic form, a Hamiltonian system of a time-dependent Tonelli
Hamiltonian function possesses infinitely many brake orbits. More precisely, by
applying Legendre transform we show that there are infinitely many symmetric
orbits of the dual Euler-Lagrange system on the configuration space. This
result contains an assertion for the existence of infinitely many symmetric
orbits of Tonelli Euler-Lagrange systems given by G. Lu at the end of [Lu09a,
Remark 6.1]. In this paper, we will present a complete proof of this assertion.
