## 1 Introduction
In particular, a comparison with the human driver is useful, on the one hand, to learn from the human information processing chain and to imitate or improve suitable elements, and on the other hand, to be able to argue the suitability of the system for public road traffic compared to a human driver to an approval authority. Accident statistics show that accidents are caused by typical driving errors, the majority of which can be divided into three categories:  
1. Errors of perception (as a rule, a road user was not seen). 2. Errors of judgment (the prediction of the situation and the behavior of others or one's own movement does not correspond to reality). 3. Manual driving errors (the vehicle is not operated adequately). Extensive technical systems have been developed in the past to compensate for these driving errors, particularly for type 3. These include ESP, ABS, emergency brake assist systems, etc. For the design of a technical system to solve the driving tasks of a human driver, this results in a sequence of steps that include the following:  
1. Object/environment perception. 2. Scene formation. 3. Prediction. 4. Generation and evaluation of alternative actions. 5. Action determination. 6. Conversion of action into a movement. 7. Re-evaluation of the influence of one's own behavior on the environment. The approval of an AD system for safe and, at the same time, economically attractive use in public road traffic must satisfy the following several aspects:  
1. The system must represent a real benefit or safety gain, according to the state of the art. For RB, this must be achieved in the best possible way in accordance with RB/GF182. 2. The system must be approved by the authorities. To this end, there must be comprehensible and officially recognized evidence and reasoning that justifies the safety and regulatory compliance of the system and withstands a legal review of the engineering duty of care at all times. 3. The system must be performant. The system must enable the transportation of persons or goods in road traffic in such a way that the legitimate expectations of users and operators are met. The system must be able to move in road traffic without presenting an unacceptably low benefit or even an unreasonable obstacle for users and other road users. To achieve the legal and regulatory requirements, these are analyzed and systematically included as mandatory requirements in the specification of the system at the beginning of the design phase. The approval regulations that are valid at the time are essential for official approval. Their compliance must be demonstrated through suitable activities. As things stand today, a sufficiently positive risk-benefit balance and proof of due diligence in the design and development of the system will be key points. In general, the duty of care includes consideration of the state of the art in science and technology, consideration of the safety generally expected by the public, consideration of reasonably foreseeable misuse, conformity of the product, and compliance with or exceeding relevant norms and standards. For a pure black-box verification of this rate on the basis of endurance driving, driving performance in the range of 6 \(\times\) 10\({}^{7}\) h would be required. System probability and completeness are necessary for release processes. Therefore, in this article, we introduce a fully provable and complete dynamical system for a traffic node based on the Euler-Lagrange equations. A main prerequisite for the Euler-Lagrange equations for AD systems is proof of a sufficiently low risk in relation to the limited release. The fixed threshold value of \(5\times 10^{-8}\) h\({}^{-1}\) for the occurrence of an accident with the damage class S1-3 serves as the starting point. The task is therefore to develop a method, approximating the Euler-Lagrange system or procedure, with which the residual risk that the occurrence rate does not meet the required verification target can be estimated on the basis of significantly fewer endurance driving hours.