Barely three months into the new year and we are happy to announce a monumental milestone reached - 150 million downloads.
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This achievement solidifies IntechOpen’s place as a pioneer in Open Access publishing and the home to some of the most relevant scientific research available through Open Access.
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We are so proud to have worked with so many bright minds throughout the years who have helped us spread knowledge through the power of Open Access and we look forward to continuing to support some of the greatest thinkers of our day.
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Thank you for making IntechOpen your place of learning, sharing, and discovery, and here’s to 150 million more!
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\n'}],latestNews:[{slug:"intechopen-signs-new-contract-with-cepiec-china-for-distribution-of-open-access-books-20210319",title:"IntechOpen Signs New Contract with CEPIEC, China for Distribution of Open Access Books"},{slug:"150-million-downloads-and-counting-20210316",title:"150 Million Downloads and Counting"},{slug:"intechopen-secures-indefinite-content-preservation-with-clockss-20210309",title:"IntechOpen Secures Indefinite Content Preservation with CLOCKSS"},{slug:"intechopen-expands-to-all-global-amazon-channels-with-full-catalog-of-books-20210308",title:"IntechOpen Expands to All Global Amazon Channels with Full Catalog of Books"},{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"}]},book:{item:{type:"book",id:"3560",leadTitle:null,fullTitle:"Advances in Landscape Architecture",title:"Advances in Landscape Architecture",subtitle:null,reviewType:"peer-reviewed",abstract:"Landscape architecture is the design of outdoor and public spaces to achieve environmental, socio-behavioral, and/or aesthetic outcomes. It involves the systematic investigation of existing social, ecological, and geological conditions and processes in the landscape, and the design of interventions that will produce the desired outcome. The scope of the profession includes: urban design; site planning; town or urban planning; environmental restoration; parks and recreation planning; visual resource management; green infrastructure planning and provision; and private estate and residence landscape master planning and design - all at varying scales of design, planning and management. This book contains chapters on recent developments in studies of landscape architecture. For this reason I believe the book would be useful to the relevant professional disciplines.",isbn:null,printIsbn:"978-953-51-1167-2",pdfIsbn:"978-953-51-5381-8",doi:"10.5772/51738",price:169,priceEur:185,priceUsd:219,slug:"advances-in-landscape-architecture",numberOfPages:940,isOpenForSubmission:!1,isInWos:1,hash:"a20614517ec5f7e91188fe8e42832138",bookSignature:"Murat Özyavuz",publishedDate:"July 1st 2013",coverURL:"https://cdn.intechopen.com/books/images_new/3560.jpg",numberOfDownloads:216410,numberOfWosCitations:180,numberOfCrossrefCitations:110,numberOfDimensionsCitations:222,hasAltmetrics:1,numberOfTotalCitations:512,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"July 20th 2012",dateEndSecondStepPublish:"September 14th 2012",dateEndThirdStepPublish:"December 28th 2012",dateEndFourthStepPublish:"February 12th 2013",dateEndFifthStepPublish:"March 28th 2013",currentStepOfPublishingProcess:5,indexedIn:"1,2,3,4,5,6",editedByType:"Edited by",kuFlag:!1,editors:[{id:"93073",title:"Dr.",name:"Murat",middleName:null,surname:"Ozyavuz",slug:"murat-ozyavuz",fullName:"Murat Ozyavuz",profilePictureURL:"https://mts.intechopen.com/storage/users/93073/images/549_n.jpg",biography:"Assoc. Prof. Dr. Murat Ozyavuz was born in 1976 in Turkey and lived there until coming to Ankara (Turkey) in 1999. He studied Landscape Architecture at the Ankara University and obtained B.S. degree in 1999. He obtained his M.S. and Ph.D. degrees from the Landscape Architecture Department of Institute of Natural and Applied Sciences in 2003 and July 2008, respectively (M. S. Thesis, Arboretum Planning Principles and Thrace University Güllapoğlu Arboretum Landscape Planning Studies, Ph.D. thesis, Planning of İğneada – Demirköy Part of Yildiz Mountains as a Biosphere Reserve). During his Ph.D. studies, he mostly worked on Landscape Planning, Protected Areas, Geographic Information Systems and Remote Sensing. Now, Dr. Ozyavuz is an Associate Professor at the Department of Landscape Architecture, Faculty of Fine Arts, Design and Architect, Namık Kemal University. 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1. Introduction
Cysts of the maxillary sinus are detected primarily as incidental findings on radiographs. These cysts often appear as rounded, dome-shaped, soft tissue masses, which are usually located on the floor of the maxillary sinus. Ectopic teeth in the maxillary sinus are readily diagnosed radiographically because they are radiopaque. Water’s view, panoramic radiography and plain skull radiography are simple and inexpensive methods that can be used in daily practice. The shape and extent of the cysts can vary widely, and the position of ectopic teeth may be found very close to the eye; in such situations, conventional radiographs may not be sufficient for determining their dimensions or relationship with anatomical structures. Thus, computed tomography (CT) should be used, and patient observation should be initiated using a multidisciplinary team that includes specialists from radiology, dentistry, and surgery departments.
Multidisciplinary treatment planning is an important key component in the long-term success and quality of treatment. In some cases, endodontic treatment should be performed by a dentist or endodontist before the operation, and the vitality of the teeth should be observed closely after the operation. Generally, treatment includes enucleation of the cyst and/or surgical excision, including endoscopic observation in some cases. Long-term multidisciplinary postoperative patient observation should be performed, especially in cases with high recurrence. The aim of this section is to shed new light on treatment approaches for cysts localized in the maxillary sinus.
2. Anatomy of the maxilla
The maxillae are the largest bones of the face, after the mandible. Each assists in forming the boundaries of three cavities: the roof of the mouth, the floor and lateral wall of the nose, and the floor of the orbit. They also enter into the formation of the fossae infratemporal and fossae pterygopalatine, and two fissures, the inferior orbital and pterygomaxillary. [1] The body of the maxilla is somewhat pyramidal in shape, and contains a large cavity, the maxillary sinus (antrum of Highmore). [1]
2.1. Blood supply and venous drainage of the maxillary teeth
The arteria (a.) maxillaris arises from the a. carotis externa, which supplies the maxillary teeth. The maxillary arch is supplied by a plexus of three arterial branches: the a. alveolares superiores anteriores, a. alveolares superiores medialis, and a. alveolares superiores posteriores. The a. alveolares superiores posteriores arises from the third division of the a. maxillaris before the a. maxillaris enters the fossa pterygopalatine (Figure 1). It continues on and enters the infratemporal surface of the maxilla to supply the maxillary sinus, the premolars, and the molars (Figure 1).
Figure 1.
A. maxillaries and a. alveolares superiores posteriores.
During operations performed in this area there may be spontaneous bleeding from these vessels during surgery and sometimes serious bleeding in the postoperative period after local anesthetics lose activity. This situation may put the patient in a dangerous situation at two time points: immediately after the operation, because of bleeding, and later, after the operation, because of infection of formed clots. Careful CT examinations before the operation and appropriate surgical management will help to avoid all intraoperative and post-operative bleeding complications.
The a. alveolares superiores medialis arises from the a. infraorbitalis as does the a. alveolares superiores anteriores. It arises within the infraorbital canal where it descends to supply the maxillary sinus and plexus at the level of the canine. The a. alveolares superiores anteriores also arises at the level of the middle superior alveolar artery and runs with it to supply the anterior portion of the maxillary arch, maxillary sinus, and anterior teeth.
The venous (v.) drainage of v. alveolares superiores posteriores, v. alveolares superiores medialis, and v. alveolares superiores anteriores drain into the plexus venosus pterygoideus. [3] Some of the most important points during operations in the maxillary part of the body are first, during the design of the flap, to protect the plexus venosus pterygoideus and lymphatic drainage and, second, to be gentle during the retraction of soft tissues.
2.2. Innervation of the maxilla and of the maxillary teeth
The nervus trigeminus (the fifth cranial nerve) is a mixed nerve (n.) responsible for sensation in the face and certain motor functions, such as biting and chewing. It has three major branches: the n. ophthalmicus, n. maxillaris, and the n. mandibularis. The n. ophthalmicus and n. maxillaris are purely sensory. The n. mandibularis has both sensory and motor functions (Figure 2). [1]
Figure 2.
The nervus trigeminus
The n. ophthalmicus carries sensory information from the scalp and forehead, the upper eyelid, the conjunctiva and cornea of the eye, the nose, the nasal mucosa, the frontal sinuses, and parts of the meninges.
The n. maxillaris carries sensory information from the lower eyelid and cheek, the nares and upper lip, the upper teeth and gums, the nasal mucosa, the palate and the roof of the pharynx, the maxillary, ethmoid and sphenoid sinuses, and parts of the meninges. It leaves the ganglion trigeminale between the n. ophthalmicus and the n. mandibularis lateral to the sinus cavernosus. The nerve leaves the cranium forward, through the foramen rotundum and enters the fossa pterygopalatina where it divides into three main branches: the n. zygomaticus, n. infraorbitalis, and truncus pterygopalatinus.
The n. infraorbitalis is a direct extension of the n. maxillaris. It leaves the fossa pterygopalatina and enters the orbit through the fissura orbitalis inferior, together with a. infraorbitalis. The nerve runs forward on the floor of the orbita in one fulcrum, which turns anteriorly to the canalis infraorbitalis. Going forward it leaves the maxilla through the foramen infraorbitale (Figure 3), positioned on the anterior wall of the maxilla under the sutura zygomaticomaxillaris, and gives rise to terminal branches.
Figure 3.
N.infraorbitalis, c. infraorbitalis and f.infraorbitalis
The n. alveolares superiores arises from the n. maxillaris in the fossa pterygopalatina just before n. infraorbitalis enters the orbita or arises from the n. infraorbitalis in the sulcus infraorbitalis. The upper alveolar nerves are divided in three groups: the n. alveolaris superior posterior, the n. alveolaris superior medius, and the n. alveolaris superior anterior. [4] Working 5 mm above the roots of the teeth in the maxilla will avoid damage to the neurovascular plexus. [5] This is one of the most important points during surgical procedures performed in the maxillary sinuses when the teeth are vital (Figure 4). A second important point is to avoid damage to the n. infraorbitalis, which is commonly damaged during elevation and retraction of mucoperiosteal flaps.
Figure 4.
N. infraorbitalis, c. infraorbitalis and f. infraorbitalis (part of figure adapted from Netter).
The n. mandibularis carries sensory information from the lower lip, the lower teeth and gums, the chin and jaw, parts of the external ear, and parts of the meninges. The mandibular nerve carries touch/position and pain/temperature sensation from the mouth. It does not carry the taste sensation; the chorda tympani is responsible for taste. However, one of its branches, the lingual nerve, carries somatic sensation from the tongue.
3. History and anatomy of the maxillary sinus
The maxillary sinus was first discovered and illustrated by Leonardo da Vinci (1452-1519), but the earliest scientific description is attributed to the British surgeon Nathaniel Highmore (1613-1685). [2] The sinus maxillaris is located behind the anterior wall of the os. maxillaris, under the orbital cavities and above the alveolar bone of the teeth. It has the shape of a pyramid, with a volume of ~15 cc, inferosuperior length of 33 mm, a mediolateral length of 23 mm, and anteroposterior length of 34 mm (Figure 5). The deepest point of the maxillary sinus is normally located in the area of the molar roots; the next deepest area is at the premolar roots. Thus, the risk of exposing the maxillary sinus intraoperatively is greatest when molar teeth are extracted (Figure 6). [6-10]
Kiliç et al. [8] evaluated the maxillary sinus regions from 92 patients, using dental cone-beam CT. This study showed that ~10.5% of molar roots were located in the maxillary sinus. Jung and Cho [9] in their study showed that 28.1% of molar roots were located in the maxillary sinus. Hirata et al. [10] investigated 1337 patients after 2038 molar extractions and found 3.8% maxillary sinus perforations. In addition to the relationship between the roots and the maxillary sinus floor, exposing the maxillary sinus intraoperatively when molar teeth are extracted depends on the shape and distance of the roots from each other, extraction technique, skill and experience. Knowledge of the anatomical shape of the maxillary sinus and the relationship between the sinus floor and tooth apices, careful planning, and good extraction technique will avoid maxillary sinus perforations.
The paranasal sinuses and the majority of the nasal cavity itself are lined with pseudostratified columnar ciliated epithelium (respiratory type). The cilia suspend a mucous blanket, which is secreted by goblet cells in the mucous membrane. [11] The sinus maxillaris has an opening for drainage, the ostium, located on the medial wall into the hiatus semilunaris. The position of the ostium does not help draining of sinus contents when the head is in an upright position. Before operation, this opening should be checked in CT scans for any obstruction. Ordinarily, it has a diameter of 5-7 mm and a length of 1-2 mm.
Figure 5.
The sinus maxillaries. Volume ~15 cc and pyramidal shape. 1. inferosuperior length 33 mm, 2. Mediolateral length 23 mm, 3. Anteroposterior length 34 mm.
Figure 6.
Maxillary sinus exposed after second molar tooth extraction.
Janner et al. [12] reported that the thickness of the Schneiderian membrane showed a wide range, with a minimum value of 0.16 mm and a maximum value of 3.461 mm. The highest mean values, ranging from 2.16 to 3.11 mm, were found for the mucosa located in the mid-sagittal regions of the maxillary sinus. Dagassan-Berndt et al. [13] stated that in the molar regions with periodontal destruction, Schneiderian membrane thickening occurred, particularly in combination with small bone layers above the root tips or periapical lesions.
4. Etiology of cysts localized in the maxillary sinus
A cyst is a lesion consisting of an epithelial sac, filled with fluid or semisolid material, and is surrounded by a connective tissue capsule. Cysts are more commonly seen in the maxilla than the mandible. The most common causes of cysts localized in the maxillary sinus are chronic infection, allergic sinonasal disease, trauma, previous surgery, obstruction of the sinus ostium, accumulations of secretions, ectopic teeth, foreign bodies (e.g., dental implant, tooth roots, graft materials), dental infections, incomplete sealing of all communications between the root canal system and periradicular tissues during endodontic treatment, mechanical obstruction of mucociliary flow, defects in ciliary capabilities to propel the mucous blanket and genetic factors. Pathologically, a cyst can develop and grow in the sinus until it reaches a large size with no serious complaint by the patient because of the anatomy of the sinus. [14-16]
5. Cyst treatment options
Marsupialization (Partsch I) procedure is a technique for making a surgical incision in the cyst capsule, minimizing intracystic pressure, and evacuating its contents, then suturing the edges of the cyst to the healthy surface of the oral mucosa to establish a large permanent opening (Figure 7).[17]
Decompression is a technique that relieves the pressure within the cyst by making a small opening in the cyst and keeping it open. This can be achieved with a drain or obturator. Each day the obturator should be removed and the cyst cavity should be irrigated. The cyst size will decrease and any damage to important structures upon enucleation will be diminished (Figures 8, 9). [18]
The decompression and marsupialization of cysts were first suggested by Partsch in the German literature in the late 19th century. [18] Indications for such marsupialization and decompression are large cysts with thin bony walls that may cause spontaneous fracture, cysts that are very close to structures such as the n. alveolaris inferior or nasal floor, and infected cysts. [17, 18, 19]
Enucleation (Partsch II procedure) is a procedure in which all pathological tissue is removed and wound edges are closed (Figure 10). [18]
Figure 7.
Marsupialization (Partsch I) procedure of a ranula.
Figure 8.
Decompression technique for a dentigerous cyst.
Figure 9.
Enucleation after 12 months of decompression.
Figure 10.
Enucleation (Partsch II procedure) of an odontogenic keratocyst.
6. Cases
6.1. Case 1
A 19-year-old male was referred to our department of oral and maxillofacial surgery with a 3-week history of swelling on the right side of the face. There was no history of trauma, pain, paresis, paresthesia, or lymphadenopathy. There was slight but obvious facial asymmetry caused by the swelling over the right maxillary region. The mass was firm and non-tender on palpation and not adherent to the overlying skin. Intra-oral examination showed little expansion of the upper right third molar region.
A panoramic radiograph and CT showed a well-defined unilocular radiolucency involving the right maxillary sinus along with the impacted third molar in the top part of the maxillary sinus in conjunction with the orbital floor. A vitality test was performed; all the teeth were vital.
Figure 11.
Panoramic radiograph and CT views.
In cases like this, careful examination is important. Points that should be checked include possible obstruction of the sinus ostium (Figures11, 12), the need for an antrostomy procedure, the route of the canalis nasolacrimalis, resorption of the posterior bony wall of the sinus, continuity of the orbital floor, and eye examinations before and after the operation. In such operations where visualization is a problem, an endoscopic-assisted approach is mandatory. [20] Endoscopic sinus surgery has been performed for various indications in maxillofacial surgical practice. It has been used for the assessment of antral pathologies, removal of foreign bodies, orthognathic procedures, and the treatment of facial fractures. [21, 22] Especially after finishing the removal of a cyst and tooth in operations like this using endoscopy, the surgeon should check any remaining pathology of the cyst, continuity of the orbital floor, to assess the possibility of a blow-out fracture [23], root tips of the teeth and any possible damage, check the aperture of sulcus nasolacrimalis, and perform an antrostomy using endoscopic assistance as needed. [24]
The operation was performed under general anesthesia, combined with local anesthesia. A mucoperiosteal flap was opened in two layers (Figure 13). Through a modified Caldwell-Luc approach, the cyst was exposed (Figure 14). At this stage in the operation, saving teeth vitality is the most important point, so it is important to work at least 5 mm away from the teeth apices. After the pus was removed from the cyst, the tooth was carefully extracted under endoscopic assistance and the remaining part of the cyst was then enucleated (Figure 15, 16). Using endoscopy, the cavity was checked (Figure 17) and packing of the sinus (Figure 18) was performed; this was removed 3 days later. Vitality of the teeth were checked for the last time at 3 months after the operation and all teeth were vital.
Figure 12.
Panoramic radiograph, lateral cephalometric and CT views showing tooth in the right maxillary sinus.
An intranasal inferior meatal antrostomy is a process of making an opening in the nasoantral wall of the inferior meatus by an intranasal route. The nasoantral wall of the inferior meatus is perforated with a curved hemostat and then this opening is enlarged. The opening should be 1.5-2 cm in diameter and as close to the floor of nose as possible (Figure 19). Intrasinus pus/debris should be removed by suction. Packing into the sinus is achieved from the posterior, packing layer-by-layer upwards and forwards (to facilitate removal through the antrostomy) and nose packing may also be required if there is severe bleeding (Figure 20). [25]
Figure 19.
Intranasal inferior meatal antrostomy technique.
Figure 20.
Packing of the sinus maxillaris.
6.2. Case 2
During an incidental radiological exam (orthopantomograph, OPG, Figure 21 at our department of radiology, a 51-year-old male with no complaints and no history of any trauma, pain, paresis, paresthesia, or lymphadenopathy, was discovered to have an ectopic tooth in the right maxillary antrum. A CT scan (Figure 21) was performed and all important points were checked. An operation was performed under general anesthesia with combined local anesthesia. Using a crestal incision, a trapezoid mucoperiosteal flap was designed and carefully elevated. The tooth was extracted under endoscopic assistance (Figures 22, 23) and a dentigerous cyst was enucleated (Figure 24). Using endoscopy, all cavities were checked and packing of the sinus was performed; this was then removed 3 days later.
Figure 21.
OPG and CT views.
Figure 22.
Extraction under endoscopic assistance.
Figure 23.
Extraction under endoscopic assistance.
Figure 24.
Enucleated dentigerous cyst.
6.3. Case 3
A 23-year-old female was referred to our department of oral and maxillofacial surgery with headache, fatigue, and difficulty in nasal breathing. After a radiological examination (figure 25), it was seen that there was an ectopic third molar in the left sinus and a wisdom lower left third molar in conjunction with a cyst. A CT scan (Figure 26) was performed and all important points were checked and it was seen that the third molar in the maxillary sinus was associated with a cyst that had occupied over two-thirds of the left maxillary sinus.
An operation was performed under general anesthesia with combined local anesthesia. A trapezoid mucoperiosteal flap was designed and carefully elevated (Figure 27). The tooth was extracted (Figure 28) and the cyst was enucleated (Figure 29). An intranasal inferior meatal antrostomy was performed with packing of the sinus (Figure 30); this was removed 3 days later. After the operation, all symptoms of headache, fatigue, and difficulty in nasal breathing resolved. Pathological specimens were sent for examination and the report showed that the cyst was an orthokeratotic odontogenic cyst.
During the operation, using envelope incision, the lower-left-third molar was also extracted and its cyst was enucleated (Figure 31). The pathological report for the lower cyst showed it to be dentigerous. The patient remains under observation (Figure 32).
Figure 25.
OPG view.
Figure 26.
CT views.
Figure 27.
Mucoperiosteal flap preparation.
Figure 28.
Tooth extraction.
Figure 29.
Enucleated cyst and extracted tooth.
Figure 30.
Intranasal inferior meatal antrostomy and packing of the sinus maxillaris.
Figure 31.
Extracted lower tooth.
Figure 32.
5 year postoperative OPG.
6.4. Case 4
A 26-year-old female was referred to our department with severe headache, fatigue, difficulty in nasal breathing, halitosis, and a slowly growing facial deformity on the right site of the face. After radiological assessment (OPG and CT) and clinical examination (Figure 33), it was seen that there was a cyst that almost completely filled the right maxillary sinus and there was slight, but obvious, facial asymmetry caused by the swelling over the right maxillary region. The mass was firm and non-tender on palpation and not adherent to the overlying skin. Intra-oral examination showed expansion of the upper right molar region.
Endodontic treatment was performed for the upper right first and second premolars and for the second molar tooth. It was decided that the first molar roots and third molars would be extracted. All important anatomical points were carefully checked on CT scans.
An operation was performed under general anesthesia, with combined local anesthesia. A trapezoid mucoperiosteal flap was designed and carefully elevated (Figure 34). The teeth were extracted and the cyst was enucleated (Figure 35); and apical resection was performed (Figure 36). An intranasal inferior meatal antrostomy was performed as was packing of the sinus (Figures 37, 38); this was removed 3 days later. Endoscopic assistance was not used because of good visualization. Pathological specimens were sent for examination and report confirmed a radicular cyst (Figure 39).
Figure 33.
Radiological (OPG and CT) and intraoral clinical views.
Figure 34.
Mucoperiosteal elevation and cyst exposure.
Figure 35.
Cyst enucleation.
Figure 36.
Apical resection.
Figure 37.
Intranasal inferior meatal antrostomy and packing of the sinus maxillaris.
Figure 38.
Closed mucoperiosteal flap and enucleated cyst.
Figure 39.
5 year postoperative OPG
6.5. Case 5
A 33-year-old male was referred to our department with difficulty in nasal breathing, halitosis, and a slowly growing facial deformity on the left side of the face. From radiological (OPG and CT) and clinical examinations (Figure 42), it was seen that there was a cyst, which was related to the left maxillary teeth and nasal floor, uplifting the sinus floor, although it did not obliterate the sinus ostium and the sinus mucosa was not infected. This information was important for the decision as to whether to perform an intranasal inferior meatal antrostomy or not. In this case, because the parameters for maxillary sinus ventilation were ideal, we did not perform an antrostomy. Before the operation all teeth on the left site underwent endodontic treatment (Figure 41).
An operation was performed under general anesthesia combined with local anesthesia. A trapezoid mucoperiosteal flap was designed and carefully elevated (Figure 42). The buccal cortex was decorticated using a round bur without damaging the cyst wall (Figure 43). Intracystic liquid was aspirated with a 20-cc syringe (Figure 44). During aspiration, cholesterol crystals were observed clearly in the cyst fluid. After enucleation of cyst (Figure 45), apical resection plus MTA retrograde filling was performed and the mucoperiosteal flap was sutured. Pathological specimens were sent for examination and the report was a radicular cyst.
Endodontic surgery involves a combination of curettage of infected tissue and removal of an infected or damaged root apex. Among the causes of failure in endodontic surgery, the most frequent is the incomplete sealing of all communications between the root canal system and periradicular tissues. Many studies have shown that bacteria that remain in the root canal system have access to the periradicular tissues after resection [26, 27] The main purpose of the root-end filling material is to provide an apical seal that prevents the movement of bacteria from the root canal system into the periapical tissues. Recently, an experimental substance, MTA, was suggested as a potential root-end filling material. In a series of in vitro studies, the sealing ability of MTA was evaluated, compared with commonly used root-end-filling materials. It was shown that MTA had significantly less dye and bacterial leakage. [26, 28, 29] Some studies have shown that when the root canal is confined hermetically and an adequate retrograde cavity depth is prepared, then variation in the root-end cutting angle does not necessarily cause any difference in microleakage. [26]
Figure 40.
Radiological (OPG and CT) examination.
Figure 41.
Endodontic treatment of the teeth.
Figure 42.
Preparation and elevation of the mucoperiosteal flap.
Figure 43.
Buccal cortex decortication.
Figure 44.
Intracystic liquid aspiration with a 20-cc syringe and view of cholesterol crystals.
Figure 45.
Enucleation of the cyst.
7. Conclusions
Because of the anatomy of the maxillary sinus, pathological structures that develop and grow in the sinus can reach a large size without any serious complaint by the patient, making early diagnosis unlikely and treatment is important to address morbidity.
OPG and CT scans aid in the diagnosis of cysts localized in the sinus and CT scans are indispensable in surgical strategy planning. Good visualization during the operation provides a better opportunity for good cleaning of pathologies, better bleeding control, and minimal trauma. For better visualization, the use of endoscopy during the operation can simplify and enhance the procedure. Also, close observation of patients in the early postoperative period and regular follow-up in the later postoperative period are important.
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Introduction",level:"1"},{id:"sec_2",title:"2. Anatomy of the maxilla",level:"1"},{id:"sec_2_2",title:"2.1. Blood supply and venous drainage of the maxillary teeth",level:"2"},{id:"sec_3_2",title:"2.2. Innervation of the maxilla and of the maxillary teeth",level:"2"},{id:"sec_5",title:"3. History and anatomy of the maxillary sinus",level:"1"},{id:"sec_6",title:"4. Etiology of cysts localized in the maxillary sinus",level:"1"},{id:"sec_7",title:"5. Cyst treatment options",level:"1"},{id:"sec_8",title:"6. Cases",level:"1"},{id:"sec_8_2",title:"6.1. Case 1",level:"2"},{id:"sec_8_3",title:"6.1.1. Intranasal inferior meatal antrostomy technique",level:"3"},{id:"sec_10_2",title:"6.2. Case 2",level:"2"},{id:"sec_11_2",title:"6.3. Case 3",level:"2"},{id:"sec_12_2",title:"6.4. Case 4",level:"2"},{id:"sec_13_2",title:"6.5. Case 5",level:"2"},{id:"sec_15",title:"7. 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Characteristics and dimensions of the Schneiderian membrane: a radiographic analysis using cone beam computed tomography in patients referred for dental implant surgery in the posterior maxilla. Clin Oral Implants Res. 2011 Dec;22(12):1446-53.'},{id:"B13",body:'Dagassan-Berndt DC, Zitzmann NU, Lambrecht JT, Weiger R, Walter C. Is the Schneiderian membrane thickness affected by periodontal disease? A cone beam computed tomography-based extended case series. J Int Acad Periodontol. 2013 Jul;15(3):75-82.'},{id:"B14",body:'Han MH, Chang KH, Lee CH, Na DG, Yeon KM, Han MC. Cystic expansile masses of the maxilla: differential diagnosis with CT and MR. AJNR Am J Neuroradiol. 1995 Feb;16(2):333-8.'},{id:"B15",body:'Caylakli F, Yavuz H, Cagici AC, Ozluoglu LN. Endoscopic sinus surgery for maxillary sinus mucoceles. Head Face Med. 2006 Sep 6;2:29.'},{id:"B16",body:'Buyukkurt MC, Omezli MM, Miloglu O. Dentigerous cyst associated with an ectopic tooth in the maxillary sinus: a report of 3 cases and review of the literature. Oral Surg Oral Med Oral Pathol Oral Radiol Endod. 2010 Jan;109(1):67-71.'},{id:"B17",body:'Scott A. Martin. Conventional Endodontic Therapy of Upper Central Incisor Combined with Cyst Decompression: A Case Report. JOE 33(6), 2007, 753-757'},{id:"B18",body:'Pogrel M A. Treatment of Keratocysts: The Case for Decompression and Marsupialization. J Oral Maxillofac Surg 63:1667-1673, 2005'},{id:"B19",body:'Miloro M. Peterson’s Principle of Oral and Maxillofacial Surgery. Second Edition.. In Miloro M., Ghali G. E., Larsen P. E., Waite P. Editors. BC Decker Inc. 2004 p.575-596'},{id:"B20",body:'Bello SA, Oketade IO, Osunde OD. Ectopic 3rd molar tooth in the maxillary antrum. Case Rep Dent. 2014;2014:620741.'},{id:"B21",body:'Varol A, Türker N, Göker K, Basa S. Endoscopic retrieval of dental implants from the maxillary sinus. Int J Oral Maxillofac Implants. 2006 ;21(5):801-4'},{id:"B22",body:'Kim JW, Lee CH, Kwon TK, Kim DK. Endoscopic removal of a dental implant through a middle meatal antrostomy. Br J Oral Maxillofac Surg. 2007, 45(5):408-9.'},{id:"B23",body:'Chen CT, Chen YR. Endoscopically assisted repair of orbital floor fractures. Plast Reconstr Surg. 2001 Dec;108(7):2011-8'},{id:"B24",body:'Nour YA. Variable extent of nasoantral window for resection of antrochoanal polyp: selection of the optimum endoscopic approach. Eur Arch Otorhinolaryngol. 2014 Jul 11.'},{id:"B25",body:'Dhingra PL, Dhingra S. Diseases of Ear, Nose and Throat. Chapter 82 5th edition, Elsevier 2010'},{id:"B26",body:'Garip H, Garip Y, Oruçoğlu H, Hatipoğlu S. Effect of the angle of apical resection on apical leakage, measured with a computerized fluid filtration device. Oral Surg Oral Med Oral Pathol Oral Radiol Endod. 2011 Mar;111(3):e50-5.'},{id:"B27",body:'Lin LM, Pascon EA, Skribner J, Langleland K. Clinical, radiographic, and histologic study of endodontic treatment failures. Oral Surg Oral Med Oral Pathol 1991;71:603-11.'},{id:"B28",body:'Torabinejad M, Watson TF, Pitt FTR. Sealing ability of a mineral trioxide aggregate when used as a root end filling material. J Endod 1993;19:591-5.'},{id:"B29",body:'Torabinejad M, Higa RK, McKendry DJ, Pitt FTR. Dye leakage of four root end filling materials: effects of blood contamination.J Endod 1994;20:159-63.'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Hasan Garip",address:"hasangarip@yahoo.com",affiliation:'
Marmara University, Faculty of Dentistry, Department of Oral and Maxillofacial Surgery, Istanbul, Turkey
Marmara University, Faculty of Dentistry, Department of Oral and Maxillofacial Surgery, Istanbul, Turkey
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1. Introduction
It was well known that the Lagrangian possesses the nonuniqueness property. It means that the constant can be added or multiplied into the Lagrangian: LNẋx→αLNẋx+β
Furthermore, the total derivative term can also be added to the Lagrangian without alternating the equation of motion: LNẋx→αLNẋx+β+df/dt,
where =fxt. This fact can be seen immediately from the variational principle with the action functional:
Sx=∫0TdtαLNẋx+β+dfdt=∫0TdtαLNẋx+β+fT−f0E1
Obviously, the last two terms contribute only at the boundary. Then the variationx→x+δx on the action and δSx=0, with conditions δx0=δxT=0, give us the same Euler-Lagrange equation:
∂∂xLNẋx−ddt∂∂ẋLNẋx=0E2
The standard Lagrangian takes the form
LNẋx=Tẋ−VxE3
where Tẋ is the kinetic energy and Vx is the potential energy of the system. For a system with one degree of freedom, the kinetic energy is Tẋ=mẋ2/2. The equation of motion associated with the Lagrangian Eq. (1) is
x¨=−1/mdVx/dx=defQE4
Recently, it has been found that actually there is an alternative form of the Lagrangian called the multiplicative form [1, 2, 3]: Lẋx=FẋGx, where F and G are to be determined. Putting this new Lagrangian into the Euler–Lagrange Eq. (2), we obtained
Lλẋx=mλ2e−Exẋmλ2+ẋλ2∫0ẋe−Exq̇mλ2dq̇E5
where Exẋ=mẋ2/2+Vx is the energy function and mλ2 is in the energy unit. We find that under the limit λ which is very large limλ→∞Lλẋx−mλ2=LNẋx, we recover the standard Lagrangian. The derivation of Eq. (5) can be found in the Appendix. Interestingly, this new Lagrangian can be treated as a generating function producing an infinite hierarchy of the Lagrangian:
Lλẋx=∑j=0∞1j!−1mλ2j−1LjẋxE6
where
Ljẋx=∑k=0jj!Tj−kVkj−k!k!2j−2k+1E7
These new Lagrangians Ljẋx, however, produce the same equation of motion. Equations (6) and (7) provide an alternative way to modify the Lagrangian Eq. (3).
The problem studied in [1, 2, 3] that is actually related to the inverse problem of calculus of variations in the one-dimensional case. The well-known result can be dated back to the work of Sonin [4] and Douglas [5].
Theorem (Sonin): For every function Q, there exists a solution gL of the equation:
gQ−x¨=∂∂xLẋx−ddt∂∂ẋLẋx.whereg=∂2∂ẋ2Lẋx≠0E8
What we did in [1, 2, 3] is that we went further to show that actually Eq. (8) admits infinite solutions.
In the present chapter, we will construct the Hamiltonian hierarchy for the system with one degree of freedom. In Section 2, the multiplicative Hamiltonian will be solved directly from Hamilton’s equations. In Section 3, the physical meaning of the parameter λ will be discussed. In Section 4, the redundancy of the Hamiltonians and Lagrangians will be explained. In the last section, a summary will be delivered.
2. The multiplicative Hamiltonian
To obtain the Hamiltonian, we may use the Legendre transformation:
HNpx=pẋ−LNẋxE9
where p=∂L/∂ẋ=mẋ is the momentum variable. The standard form of the Hamiltonian is
HNpx=p22m+VxE10
which is nothing but the total energy of the system. The action is then
Spx=∫0Tdtpẋ−HNpxE11
With the variations x→x+δx and p→p+δp, with conditions δx0=δxT=0, the least action principle δSx=0 gives us
ẋ=∂∂pHNpx, ṗ=−∂∂xHNpxE12
which are known as a set of Hamilton’s equations.
We now introduce a new Hamiltonian, called the multiplicative Hamiltonian, in a form
Hpx=KpWx,E13
where Kp and Wx are to be determined. Equations (12) and (3) give us a new equation:
0=1m∂∂xHNpx+ṗ∂2∂p2HNpx+pm∂2∂x∂pHNpxE14
Replacing HN by H and inserting Eqs. (13) into (14), we obtain
0=d2Kdp2+1mṗWdWdxpdKdp+KE15
Now we define
A=def1mṗWdWdxE16
where A is a constant to be determined. Equation (16) can be immediately solved and result in
Wx=ae−mAVxE17
where a is a constant of integration. Substituting Eq. (17) into Eq. (15), we find that the function Kp is in the form
Kp=be−Ap22E18
where b is another constant. Then the multiplicative Hamiltonian Eq. (13) becomes
Hpx=ce−Ap22−mAVxE19
where c=ab. If we now choose c=−mλ2 and A=1/m2λ2, the Hamiltonian Eq. (19) becomes
Furthermore, we can rewrite the multiplicative Hamiltonian Eq. (20) in terms of the series:
Hλpx=∑j=0∞1j!−1mλ2j−1HjpxE26
whereHjpx≡HNj=p2/2m+Vxj. It is not difficult to see that Hjpx produces exactly the equation of motion Eq. (21).
From the structure of Eqs. (26) and (6), it must be a hierarchy of the Legendre transformation. To establish such hierarchy, we start to rewrite the momentum Eq. (24) in the form
Eq. (32) can be considered as the modified Hamilton’s equations for each Hjpx in the hierarchy. Obviously, for j=1, we retrieve the standard Hamilton’s Eq. (12), since p1=p=mẋ.
From the structure of the multiplicative Hamiltonian Eq. (20), it seems to suggest that the exponential of the function, defined on phase space, is always a solution of the Eq. (14). Then we now introduce an ansatz form of the Hamiltonian as
Ha,bpx=beaZpxE33
where a and b are constants to be determined. Substituting Eq. (33) into Eq. (14), we obtain
We find that if we take HNpx=Zpx to be the standard Hamiltonian, the first three terms in Eq. (34) give us back Eq. (14). Then the last bracket must vanish and gives us an extra-relation:
0=ṗ∂HN∂p2+pm∂HN∂p∂HN∂xE35
or
0=ṗ∂HN∂p+pm∂HN∂xE36
We immediately see that actually Eq. (36) is a consequence of the conservation of the energy of the system:
0=dHNdt=∂HN∂p∂p∂t+∂HN∂x∂x∂t=ṗ∂HN∂p+pm∂HN∂xE37
Then what we have here is another equation that can be used to determine for the Hamiltonian subject to the equation of motion Eq. (21). To see this, we may start with the standard form of the Hamiltonian HNpx=Tp+Vx, where Tp is a function of the momentum and to be determined. Inserting the Hamiltonian into Eq. (36), we obtain
0=ṗdTdp+pmdVdxE38
Using Eqs. (21) and (38), it can be rewritten in the form
0=ṗdTdp−pmE39
Since ṗ≠0, it means that the term inside the bracket must be zero and
∫dT=∫pmdp→Tp=p22m+CE40
where C is a constant which can be chosen to be zero. So we successfully solved the standard Hamiltonian.
Next, we put Zpx=KpWx which is in the multiplicative form Eq. (13) into Eq. (36), and we obtain
0=WṗWdKdp+pmKdWdxE41
or
mKpdKdp=1WdWdVE42
We see that both sides of Eq. (42) are independent to each other. Then Eq. (42) holds if both sides equal to a constant β. We have now for the left-hand side
mKp∂K∂p=β
∫dKK=β∫pmdp→Kp=C1eβTpE43
where C1 is a constant to be determined. Next, we consider the right-hand side
1WdWdV=β
∫dWW=β∫dV→Wx=C2eβVxE44
where C2 is a constant to be determined. Then finally the function Zpx becomes
Zpx=C1C2eβHNpxE45
where HNpx is the standard Hamiltonian. If we now choose C1C2=−mλ2 and β=−1/mλ2, the function Zpx is exactly the same with Eq. (20).
We see that with Eq. (36) the Hamiltonian can be easily determined. Here we come with the conclusion that in every function Q′=−dVdx, there exist infinite Hamiltonians of equation
Q′−ṗ=ṗ∂H∂p+pm∂H∂xE46
The existence of solutions of Eq. (46) implies that actually we can do an inverse problem of the Hamiltonian for the systems with one degree of freedom.
Remark: The perspective on nonuniqueness of Hamiltonian, as well as Lagrangian, here in the present work is quite different from those in Aubry-Mather theory [6, 7] (see also [8]). What they had been investigating is the modification of the Tonelli LagrangianLη≔L−η̂, where mechanical Lagrangian LNẋx=Tẋ−Vx is one of Tonelli Lagrangians. Here η̂=<ηx,ẋ>:TM→R and ηx is a closed 1-from on the manifold M. This means that ∫dtLη and ∫dtL will have the same extremals and therefore the same Euler-Lagrange evolution, since δ∫dtη̂=0. Thus for a fixed L, the extreamise of the action will depend only on the de Rham cohomology class c=η∈H1MR. Then we have a family of modified Lagrangians, parameterized over H1MR. With the modified Tonelli LagrangianLη, one can easily find the associated Hamiltonian Hηxp=Hxηx+p, where the momentum is altered: p→p+ηx. Then we also have a family of modified Hamiltonians, parameterized over H1MR. To make all this more transparent, we better go with a simplest example. Consider the modified LagrangianLϵ≔LN+ϵẋ, where ϵ is a constant. We find that a new action differs from an old action by a constant depending on the endpoints, ∫abdtLϵ=∫abdtLN+ϵxb−xa, and they give exactly the same Euler-Lagrange equation (see also Eq. (1)). With this new LagrangianLϵ, we can directly obtain the Hamiltonian Hϵxp=HNxp+ϵ.
3. Harmonic oscillator
In this section, we give an explicit example, e.g., the harmonic example, and also give the physical interpretation of the parameter λ. The standard Hamiltonian for the harmonic oscillator reads
Hpx=p22m+kx22E47
Then the multiplicative Hamiltonian for the harmonic oscillator is
Hλpx=−mλ2e−1mλ2p22m+kx22E48
Now we introduce η=xp, and then we consider
dηdtλ=J∂Hλ∂ηwhere∂∂η=∂/∂x∂/∂pE49
wheretλ is a time variable associated with the multiplicative Hamiltonian and J is the symplectic matrix given by
where E=T+V is the energy function and t is the standard time variable associated with the Hamiltonian Eq. (47). Equation (51) suggests that the λ-flow is comprised of infinite different flows on the same trajectory on the phase space (see Figure 1).
Figure 1.
Differential flows on the same trajectory on the phase space.
This means that we can choose any Hamiltonian in the hierarchy to work with. The physics of the system remains the same but with a different time scale. Then we may say that the parameter λ plays a role of scaling in the Hamiltonian flow on the phase space. From Eq. (52), we see that as mλ2→∞, only the standard flow survives, and of course we retrieve the standard evolution t1=t of the system on phase space.
Next we consider the standard Lagrangian of the harmonic oscillator
LNẋx=mẋ22−kx22E53
and the multiplicative Lagrangian is
Lλẋx=mλ2e−Exẋmλ2+ẋλ2∫0ẋe−Exq̇mλ2dq̇E54
whereExẋ=mẋ2/2+kx2/2 is the energy function. We know that Lagrangian Eq. (54) can be rewritten in the form
Lλẋx=∑j=0∞1j!−1mλ2LjẋxE55
where
Ljẋx=∑k=0jj!mẋ2/2j−kkx2/2kj−k!k!2j−2k+1E56
The action of the system is given by
Sx=∫0TdtLλẋx=∑j=1∞∫0TdtjLjẋxE57
where
dtj=1j!1mλ2k−1dtE58
The variation x→x+δx with conditionsδx0=δxT=0 results in
δSx=∑j=1∞∫0Tdtj∂Lj∂x−ddtj∂Lj∂xjE59
wherexj=dx/dtj. Least action principle δSx=0 gives infinite Euler-Lagrange equations
0=∂Lj∂x−ddtj∂Lj∂xj,j=1,2,3E60
which produce the equation of motions
d2xdtj2=−kxmE61
associated with different time variables. Again in this case, we have the same structure of equation of motion for each Lagrangian in hierarchy but with a different time scale. From Eq. (58), we see that as mλ2→∞, only the standard flow survives, and of course we retrieve the standard evolution t1=t of the system. Then the parameter λ also plays the role of scaling in the Lagrangian structure.
4. Redundancy
From previous sections, we see that there are many forms of the Hamiltonian that you can work with. One may start with the assumption that any new Hamiltonian is written as a function of the standard Hamiltonian HN: H=fHN. Inserting this new Hamiltonian into Hamilton’s equations, we obtain
f′E∂H∂x=−∂p∂τ,f′E∂H∂p=∂x∂τE62
wheref′E=dfHN/HN with fixing HN=E and t=f′HNτ is the rescaling of time parameter. This result agrees with what we have in Section 3, rescaling the time evolution of the system. However, there are some major different features as follows. The first thing is that our new Hamiltonians contain a parameter λ, since the explicit forms of the Hamiltonian are obtained. With this parameter, it makes our rescaling much more interesting with the fact that the rescaling time variables depend on also the parameter (see Eq. (52)). Then it means that we know how to move from one scale to another scale and of course we know how to obtain the standard time evolution by playing with the limit of the parameter λ. Without explicit form of the new Hamiltonian, which contains a parameter, we cannot see this fine detail of family of rescaling time variables, since there is only a fixed parameter E. The second thing is that actually the new Hamiltonian Eq. (20), which is a function of the standard Hamiltonian, can be obtained from the Lagrangian Eq. (5) by means of Legendre transformation. What we have seen is that Lagrangian Eq. (5) is nontrivial and is not a function of the standard Lagrangian. Again this new Lagrangian contains a parameter λ, the same with the one in the new Hamiltonian. With this parameter, the Lagrangian hierarchy Eq. (7) is obtained. What we have here is a family of nontrivial Lagrangians to work with, producing the same equation of motion, as a consequence of nonuniqueness property. An importance thing is that there is no way you can guess the form of this family of Lagrangian without our mechanism in the appendix. This means that the Hamiltonian in the form H=fHN cannot deliver all these fine details. The explicit form of the Hamiltonian Eq. (20) allows us to study in more detail and is definitely richer than the standard one.
5. Summary
We show that actually there exist infinite Hamiltonian functions for the systems with one degree of freedom. We may conclude that there exists the reverse engineering of the calculus of variation on phase space (see Eq. (44)). Furthermore, the solution of Eq. (44) exists not only as one but infinite. Interesting fact here is that these new Hamiltonians come with the extra-parameter called λ. We give the interpretation that the term mλ2 involves the time scaling of the system. This means that we can pick any Hamiltonian or Lagrangian to study the system, but the evolution will be in different scales.
In the case of many degrees of freedom, the problem turns out to be very difficult. Even in the case of two degrees of freedom, the problem is already hard to solve from scratch. We may start with an anzast form of the Lagrangian: Lẋẏxy=FẋẏGxy. This difficulty can be seen from the fact that we have to solve a non-separable coupled equation. A mathematical trig or further assumptions might be needed for solving Fẋẏ and Gxy. The investigation is now monitored.
Furthermore, promoting the Hamiltonian Eq. (20) to be a quantum operator in the context of Schrodinger’s equation is also an interesting problem. This seems to suggest that an alternative form of the wave function for a considering system is possibly obtained. This can be seen as a result from that fact that with new Hamiltonian operator, we need to solve a different eigenvalue equation, and of course a new appropriate eigenstate is needed. From the Lagrangian point of view, extension to the quantum realm in the context of Feynman path integrals is quite natural to address. However, this problem is not easy to deal with since the Lagrangian multiplication is not in the quadratic form. Then a common procedure for deriving the propagator is no longer applicable. Further study is on our program of investigation.
Acknowledgments
The author would like to thank Kittikun Sarawuttinack, Saksilpa Srisukson, and Kittapat Ratanaphupha for their interests and involving themselves in the investigation on this topic.
Notes
The content in this chapter is collected from a series of papers [1, 2, 3].
Appendix
In this section, we will demonstrate how to solve the multiplicative Lagrangian Eq. (5). We introduce here again the Lagrangian Lẋx=FẋGx, where F and G are to be determined. Inserting the Lagrangian into the action and performing the variationx→x+δx, with conditions δx0=δxT=0, we obtain
δSx=∫0Tdt−ddtGdFdẋ+FdGdxδxE63
The least action principle states that the system will follow the path which δS=0 resulting in
Using equation of motion, we observe that the coefficient of the second term depends only x variable. Then we may set
1x¨GdGdx=defA→1GdGdx=−AmdVdxE66
We find that it is not difficult to see that the function G that satisfies Eq. (66) is
Gx=α1e−AVx/mE67
where α1 is a constant to be determined. Inserting Eq. (66) into Eq. (65), we obtain
d2Fdẋ2−AF−ẋdFdẋ=0E68
and the solution F is given by
Fẋ=α2ẋ−α3e−Aẋ2/2+ẋA∫0ẋdve−Av2/2E69
whereα2 and α3 are constants. Then the multiplicative Lagrangian is
Lẋx=k1ẋ−k2e−Aẋ2/2+ẋA∫0ẋdve−Av2/2e−AVx/mE70
wherek1=α1α2 and k2=α1α3 are new constants to be determined. We find that if we choose k1=0,A=1/λ2 which is a unit of inverse velocity squared and k2=−mλ2 which is in energy unit, Lagrangian Eq. (70) can be simplified to
limλ→∞Lλẋx−mλ2=mẋ22−Vx=LNẋxE71
the standard Lagrangian at the limit λ approaching to infinity. Therefore, the Lagrangian Eq. (70) is now written in the form
Lλẋx=mλ2e−ẋ2/2λ2+ẋλ2∫0ẋdve−v2/2λ2e−Vx/mλ2E72
which can be considered as the one-parameter extended class of the standard Lagrangian.
\n',keywords:"Hamiltonian, Lagrangian, nonuniqueness, variational principle, inverse problem of calculus of variations",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/70225.pdf",chapterXML:"https://mts.intechopen.com/source/xml/70225.xml",downloadPdfUrl:"/chapter/pdf-download/70225",previewPdfUrl:"/chapter/pdf-preview/70225",totalDownloads:266,totalViews:0,totalCrossrefCites:0,dateSubmitted:"November 6th 2018",dateReviewed:"June 14th 2019",datePrePublished:"December 13th 2019",datePublished:"June 17th 2020",dateFinished:"November 26th 2019",readingETA:"0",abstract:"The alternative Hamiltonians for systems with one degree of freedom are solved directly from the Hamilton’s equations. These new Hamiltonians produce the same equation of motion with the standard one (called the Newtonian Hamiltonian). Furthermore, new Hamiltonians come with an extra-parameter, which can be used to recover the standard Hamiltonian.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/70225",risUrl:"/chapter/ris/70225",signatures:"Sikarin Yoo-Kong",book:{id:"7788",title:"Progress in Relativity",subtitle:null,fullTitle:"Progress in Relativity",slug:"progress-in-relativity",publishedDate:"June 17th 2020",bookSignature:"Calin Gheorghe Buzea, Maricel Agop and Leo Butler",coverURL:"https://cdn.intechopen.com/books/images_new/7788.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",isbn:"978-1-78985-358-2",printIsbn:"978-1-78985-357-5",pdfIsbn:"978-1-83880-988-1",editors:[{id:"99400",title:"Dr.",name:"Calin Gheorghe",middleName:null,surname:"Buzea",slug:"calin-gheorghe-buzea",fullName:"Calin Gheorghe Buzea"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. The multiplicative Hamiltonian",level:"1"},{id:"sec_3",title:"3. Harmonic oscillator",level:"1"},{id:"sec_4",title:"4. Redundancy",level:"1"},{id:"sec_5",title:"5. Summary",level:"1"},{id:"sec_6",title:"Acknowledgments",level:"1"},{id:"sec_6",title:"Notes",level:"1"},{id:"sec_7",title:"Appendix",level:"1"}],chapterReferences:[{id:"B1",body:'Sarawuttinack K, Yoo-Kong S, Tanasittikosil M. Multiplicative form of the Lagrangian. Theoretical and Mathematical Physics. 2016;189(3):1693-1711. DOI: 10.1134/s0040577916120023'},{id:"B2",body:'Srisukson S, Sarawuttinack K, Yoo-Kong S. The multiplicative Hamiltonian and its hierarchy. Journal of Physics: Confernce Series-Siam Physics Congress 2017(SPC2017). 2018;901:012167. DOI: 10.1008/1742-6596/901/1/012167'},{id:"B3",body:'Srisukson S, Ratanaphupha K, Yoo-Kong S. Hamiltonian Zoo for the System with One Degree of Freedom. arXiv:1802.06370. 2018'},{id:"B4",body:'Sonin NJ. About determining maximal and minimal properties of plane curves (in Russian) [English translation]. Lepage Research Institute Archive. 2012;1(1–2):1-68'},{id:"B5",body:'Douglas J. Solution of the inverse problem of the calculus of variations. Transactions of the American Mathematical Society. 1941;50(1):71-128. DOI: 10.2307/1989912'},{id:"B6",body:'Aubry A, Le Daeron YP. The discrete Frenkel-Kontorova model and its extensions. Physica D: Nonlinear Phenomena. 1983;8(3):381-422'},{id:"B7",body:'Mather NJ. Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology. 1982;21(4):457-467'},{id:"B8",body:'Fathi A. Weak Kam theorem and Lagrangian. Dynamics (Pembroke, Ont.). 2007. Available from: https://www.math.u-bordeaux.fr/~pthieull/Recherche/KamFaible/Publications/Fathi2008_01.pdf'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Sikarin Yoo-Kong",address:"sikariny@nu.ac.th",affiliation:'
The Institute for Fundamental Study (IF), Naresuan University, Phitsanulok, Thailand
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Open Access publication costs can often be designated directly in the grants or in specific budgets allocated for that purpose. Many of the most important funding organisations encourage, and even request, that the projects they fund are made available at no cost to the wider public. IntechOpen strives to maintain excellent relationships with these funders and ensures compliance with mandates.
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In order to help Authors identify appropriate funding agencies and institutions, we have created a list, based on extensive research on various OA resources (including ROARMAP and SHERPA/JULIET) of organizations that have funds available. Before consulting our list we encourage you to petition your own institution or organization for Open Access funds or check the specifications of your grant with your funder to ascertain if publication costs are included. Where you are in receipt of a grant you should clarify:
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Does your institution already have a budget for covering Open Access publication costs?
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Does your grant list Open Access publication fees as legitimate direct/indirect costs?
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Please note that this list is not a definitive one and is updated regularly. To suggest possible modifications or the inclusion of your institution/funder, please contact us at oapf@intechopen.com
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