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Rationale: Wrist-hand extension function rehabilitation is a vital and difficult part of hand function recovery in spastic stroke patients. Although botulinum toxin type A (BoNTA) injection plus post injection therapy was applied to the wrist-hand rehabilitation in previous reports, conclusion was inconsistent in promoting function. For this phenomenon, proper selection of patients for BoNTA injection and correct choice of post-injection intervention could be the crucial factors for the function recovery. Patient concerns: We reported a 46-year-old male suffered a spastic hemiplegia with wrist- hand extension deficit. Diagnoses: Computed tomography showed cerebral hemorrhage in the left basal ganglia region. Interventions: Four hundred units of BoNTA were injected into the spasticity flexors, and four-week post injection surface electromyography (sEMG) biofeedback therapy was applied to the patient. Outcomes: The patient exhibited post-intervention improvement in wrist-hand extensors performance (strength, range of motion, sEMG signals), the flexors spasticity, and upper extremity function. Lessons: The present case showed that 4-week of BoNTA injection plus sEMG biofeedback exercise improved the performance and function of wrist-hand extensors in the patient for short- and long-term. Proper selection of patients for BoNTA injection and correct choice of post injection exercise could play a vital role in the hand rehabilitation for patient with spastic hemiplegia.

In this paper we study the higher-order differential variational principle and differential equations of motion for mechanical systems in event space. Based on the higher-order d'Alembert principle of the system, the higher-order velocity energy and the higher-order acceleration energy of the system in event space are defined, the higher-order d'Alembert—Lagrange principle of the system in event space is established, and the parametric forms of Euler—Lagrange, Nielsen and Appell for this principle are given. Finally, the higher-order differential equations of motion for holonomic systems in event space are obtained.

Starting from the Мещерский equations, the higher order D'Alembert-Lagrange principle for mechanical system of variable mass is obtained, and different kinds of higher order differential equations of motion for holonomic mechanical system of variable mass are derived. The result shows that these equations extend and optimize the correlative theory of holonomic mechanics.

Based on the dynamic equations of particles system in non-inertial coordinate system, the higher order differential variational principle of relative motion for mechanical system is obtained, and the energy of higher order relative velocity of the mechanical system is introduced, the different kinds of higher order dynamic equations of holonomic mechanical system in relative motion are derived, and an example is given to illustrate the application of the results.

Starting from Newtonian kinetic equations of a particle system, the energy of higher order-velocity of the system is introduced; higher-order Lagrange equations, higher-order Nielsen equations and higher-order Appell equations of a holonomic mechanical system are derived, from which we prove that the three kinds of higher-order differential equations of motion of the holonomic system are equivalent to each other. The result indicates that the higher-order differential equations of motion of the holonomic system reveal the relationship between the chan ges of the system's motion state and the rate of change of force at every order, which cannot be obtained by using Newtonian kinetic equations and the tradition al analytical mechanical equations. Therefore, the higher-order differential equ ations of motion of the holonomic system are a complement to the second-order di fferential equations of motion, including Newtonian kinetic equations and the tr aditional Lagrange equations, Nielsen equations and Appell equations.

First, the theorem on energy of higher order velocity of the holonomic potential mechanical system is presented with on explaining of the physical meaning of energy of higher order velocity of the system. The general criterion of potential mechanical system is then presented. On this basis, the higher order Lagrange function is introduced,the higher order Lagrange equations of holonomic potential mechanical system are derived, and the higher order cyclic integral and the integral of higher order generalized energy of the system are obtained.