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Equations for the synchrotron motion are derived from the Hamiltonian, which was composed of coasting, betatron and synchrotron motions. The synchrotron oscillation is not only an oscillation of the revolution frequency but also an oscillation of the average radius. The synchrotron oscillation, which is both longitudinal and horizontal oscillations, can exchange energy with the betatron oscillation, which is a horizontal oscillation. The synchrotron oscillation occurs under a constant particle velocity and the s description is equivalent to the t description. Coriolis like force acting on its horizontal oscillation brings out its longitudinal oscillation. INTRODUCTION We discuss the oscillating synchrotron motion [1] using the Hamiltonian composed of coasting, synchrotron and betatron motions. We call it the synchrotron oscillation. The Hamiltonian, which clarified the synchro-betatron resonant coupling mechanism in a storage ring, revealed that the energy exchange between the synchrotron and betatron oscillations was possible [2]. The synchrotron and betatron oscillations are obtained with s as an independent variable [3]. The betatron oscillation is an oscillation in the horizontal direction. We call it a horizontal oscillation. Since the synchrotron oscillation is accompanied by orbiting particles and occurs in the orbital direction, it is an oscillation in the longitudinal direction. We call it a longitudinal oscillation. Unless a horizontal component of the synchrotron oscillation exists, the energy exchange between those two oscillations is impossible. We show that the synchrotron oscillation derived from the Hamiltonian is not only a longitudinal oscillation but also a horizontal one and discuss about its new feature. THE HAMILTONIAN FOR ORBITING PARTICLES In the right-handed curvilinear coordinate system , , , x s z A is the vector potential, is the momentum. Here s is the orbital length. For an orbital momentum s p  , the particle is moving in a counterclockwise direction. We assume that an on-momentum particle of mass m and momentum 0 p is revolving (without oscillating motion) on the reference closed orbit of the average radius R under the dipole magnetic field 0 B  . Around the reference closed orbit, x is the horizontal coordinate and x p is the horizontal momentum. We have the following relations for the on-momentum particle: the velocity ds v dt  which satisfies , the orbit angle which satisfies s R   and the angular revolution frequency  which satisfy d dt    . Since we have d s c dt R R           , R    . (1) Here q is the elementary charge, c is the velocity of light, t is time and  is the bending radius (curvature) where 0 0 p c qB    is satisfied. We have the following relations: , and , (2) where and 0 E is the total energy of the on momentum particle. We have , and 0 0 s p p p    . We neglect vertical motion and put and . Then define and . is the momentum deviation and is the deviation of the kinetic energy of the off-momentum (orbiting) particle from the on-momentum (revolving) particle. Define a symbol of rationalized fractional deviation 2 0 E E     . We transform coordinate system of onto then to . (3) Around the off-momentum closed orbit, is the horizontal coordinate and is the horizontal momentum. is the arrival time of the off-momentum particle. They satisfy the following relations [4]: , (4) , (5)

New Feature of the Oscillating Synchrotron Motion Derived from the Hamiltonian Composed of Three Motions